A team of researchers recently published a paper claiming that large reasoning models (LRMs) are not capable of “thinking” or making logical choices using some simple logical puzzles. In the paper, LLMs (Large Language Models) and LRMs are both capable of solving the puzzles when low-complexity versions of the puzzles are presented, but when high-complexity versions are presented, both LRMs and LLMs are incapable of logic.
In this post, I’ll be referring to the Tower of Hanoi puzzle, one of the puzzles used in the paper. The rules and an explanation of the game can be found on Wikipedia.
In this post, I’ll attempt to modify the experiment design to account for two possible flaws in the paper’s methodology:
I had some personal experience with the second potential flaw when, in an attempt to prove that I am smarter than a machine, I tried to write out all of the moves necessary to solve the Tower of Hanoi with 9 rings. I ended up spending a lot of time trying to go back through moves and remember what the “current state” was. However, when I solved the puzzle in a way that I could always see the “current state”, I was a lot faster.
Okay, here we will set up some functions so that the LLM can play our game.
import numpy as np
import re
STACK_OFFSET = 3 ## An offset from the "current state" string to identify where the stacks start
STACK_LETTERS = "ABC" ## All of the letters in stacks
INCORRECT_MOVE_LANGUAGE = "Invalid move. "
def process_move(move, previous_state):
"""
Take a move an either return the state after the move was created or return None.
"""
if len(move) != 2:
return None
move_disc = move[0]
move_stack = move[1]
if not move_disc.isdigit():
return None
if move_stack not in STACK_LETTERS:
return None
previous_state_list = previous_state.split("\n")
final_discs = [previous_state_list[STACK_OFFSET + i].strip()[-1] for i in range(3)]
if move_disc not in final_discs:
return None
final_disc_stack = previous_state_list[STACK_OFFSET + STACK_LETTERS.index(move_stack)]
if final_discs[STACK_LETTERS.index(move_stack)].isdigit() and (move_disc > final_discs[STACK_LETTERS.index(move_stack)]):
return None
previous_state_list[STACK_OFFSET + final_discs.index(move_disc)] = previous_state_list[STACK_OFFSET + final_discs.index(move_disc)][:-2]
previous_state_list[STACK_OFFSET + STACK_LETTERS.index(move_stack)] += " " + move_disc
return "\n".join(previous_state_list)
def get_next_state(previous_state, response):
"""
Get the next state from a move, calling process_move. Parse the move from a LLM response, and
"""
move = re.sub("[^\w\s]", "", response).strip().split("\n")[-1].split(" ")[-1]
next_state = process_move(move, previous_state)
if next_state is not None:
return next_state.replace(INCORRECT_MOVE_LANGUAGE, "")
else:
return INCORRECT_MOVE_LANGUAGE + previous_state.replace(INCORRECT_MOVE_LANGUAGE, "")
def get_valid_moves(current_state):
"""
Get a list of possible valid moves based on the current staate of the game
"""
end_chars = [x.strip()[-1] for x in current_state.split("\n")[STACK_OFFSET:STACK_OFFSET+3]]
valid_moves = []
for i, disc in enumerate(end_chars):
if disc.isdigit():
for j in range(3):
if j == i:
continue
if end_chars[j] <= disc:
continue
valid_moves.append(disc + STACK_LETTERS[j])
return valid_moves
def get_random_move(current_state):
"""Choose a random move"""
return np.random.choice(get_valid_moves(current_state))
def get_full_stack(num_discs):
"""Get a full stack of discs together in the proper order. To be used in building a system prompt."""
full_stack = ""
for i in range(num_discs):
full_stack = " " + str(i) + full_stack
return full_stack
def get_game_start_end(num_discs):
"""
Get starting and final states of the game.
"""
full_stack = get_full_stack(num_discs)
starting_game = f"""The current state of the game is:
A | {full_stack} B | C |
ending_game = f"""The current state of the game is:
A | B | C | {full_stack}
return starting_game, ending_game
I used this chunk to debug the gameplay and demonstrate how long it takes for a model to solve the puzzle at random.
current_state, ending_state = get_game_start_end(2)
num_iterations = 0
num_discs = 3
current_state, ending_state = get_game_start_end(num_discs)
## Looking at
while current_state != ending_state:
num_iterations += 1
if num_iterations > 20:
break
move = get_random_move(current_state)
current_state = get_next_state(current_state, move)
print(current_state)
print(get_valid_moves(current_state))
print(f"Completed the game at {num_iterations} iterations")
The current state of the game is:
```
A | 2 1
B | 0
C |
```
['1C', '0A', '0C']
The current state of the game is:
```
A | 2 1 0
B |
C |
```
['0B', '0C']
The current state of the game is:
```
A | 2 1
B |
C | 0
```
['1B', '0A', '0B']
The current state of the game is:
```
A | 2 1 0
B |
C |
```
['0B', '0C']
The current state of the game is:
```
A | 2 1
B |
C | 0
```
['1B', '0A', '0B']
The current state of the game is:
```
A | 2 1
B | 0
C |
```
['1C', '0A', '0C']
The current state of the game is:
```
A | 2 1 0
B |
C |
```
['0B', '0C']
The current state of the game is:
```
A | 2 1
B | 0
C |
```
['1C', '0A', '0C']
The current state of the game is:
```
A | 2 1
B |
C | 0
```
['1B', '0A', '0B']
The current state of the game is:
```
A | 2
B | 1
C | 0
```
['1A', '0A', '0B']
The current state of the game is:
```
A | 2 0
B | 1
C |
```
['0B', '0C', '1C']
The current state of the game is:
```
A | 2 0
B |
C | 1
```
['0B', '0C', '1B']
The current state of the game is:
```
A | 2
B | 0
C | 1
```
['0A', '0C', '1A']
The current state of the game is:
```
A | 2 0
B |
C | 1
```
['0B', '0C', '1B']
The current state of the game is:
```
A | 2
B |
C | 1 0
```
['2B', '0A', '0B']
The current state of the game is:
```
A |
B | 2
C | 1 0
```
['2A', '0A', '0B']
The current state of the game is:
```
A |
B | 2 0
C | 1
```
['0A', '0C', '1A']
The current state of the game is:
```
A | 1
B | 2 0
C |
```
['1C', '0A', '0C']
The current state of the game is:
```
A | 1 0
B | 2
C |
```
['0B', '0C', '2C']
The current state of the game is:
```
A | 1
B | 2
C | 0
```
['1B', '0A', '0B']
Completed the game at 21 iterations
Now, we’re ready to set up a function that calls the LLM to play the game.
from config import OPENAI_API_KEY
import os
from openai import OpenAI
import re
import time
os.environ["OPENAI_API_KEY"] = OPENAI_API_KEY
def llm_plays_tower_hanoi(num_discs, model, restrict_token_length = False,
max_num_iterations = 50, seconds_sleep_per_step = 1, temperature = 1, starting_current_state = None):
"""Have an openAI model play the tower of hanoi
Args:
num_discs: The number of discs to include in the game
model: The model to use
restrict_token_length: Whether to restrict the model's input to only include the most recent game state. If True, the model will only receive the current state
max_num_iterations: the maximum number of iterations.
seconds_sleep_per_step: The number of seconds to sleep for each move in gameplay. Useful if you are hitting rate limits.
temperature: The model's temperature
"""
sys_prompt = f"""
You are a puzzle master completing the tower of Hanoi puzzle.
This puzzle will be represented to you in this fashion:
A | {get_full_stack(num_discs)}
B |
C |
There are three stacks: A, B, and C.
There are {num_discs} discs on the stacks, represented as numbers, with 0 being the smallest and {num_discs - 1} being the largest. Each digit is a disc
Your goal is to move all of the discs from stack A to stack C, in the proper order.
You must follow the following rules:
1. Only one disc may be moved at a time.
2. Each move consists of taking the top disc from one of the stacks and placing it on the top of another stack or on an empty stack.
3. No disc may be placed on top of a disc that is smaller than it. (so, disk 1 cannot be placed on top of disk 0)
The game will be represented to you in an intermediate state, where the right-most disc represents the "top" of each stack.
At the start of the game, the puzzle looked like:
A | {get_full_stack(num_discs)}
B |
C |
To win, you must move all the discs to stack C, so the puzzle will look like:
A |
B |
C | {get_full_stack(num_discs)}
Your role is to plan ahead and make the best move based on the current state of the puzzle, so that the puzzle moves closer to the winning state. Think step by step and when you are ready, return your move as the last line of your response.
To make a move, state the number of the disk and the stack you want to move it to in the following manner:
1B
The move `1B` moves the disc 1 to the top of stack B. The move `0C` moves the disc 0 to the top of stack C.
Think step by step about your move, considering how your move will enable future moves to reach the winning state.
When you are ready, return your chosen move as the last line of your response.
"""
print(sys_prompt)
client = OpenAI()
current_state, ending_state = get_game_start_end(num_discs)
if starting_current_state is not None:
current_state = starting_current_state
num_iterations = 0
messages = [
{"role": "system", "content": sys_prompt},
{"role": "user", "content": current_state}
]
while current_state != ending_state:
valid_moves = "\nValid Moves are: " + "[`" + "`, `".join(get_valid_moves(current_state)) + "`]"
num_iterations += 1
if num_iterations >= max_num_iterations:
break
response = client.chat.completions.create(
model=model,
temperature=temperature,
messages=messages
)
move = response.choices[0].message.content
print("-"*20)
print(current_state + valid_moves)
print("-"*20)
print(move)
messages.append({"role": "assistant", "content": move})
previous_state = current_state + valid_moves
previous_move = move
current_state = get_next_state(current_state, move)
if restrict_token_length:
## Only pass in the current state of the game
messages = [
{"role": "system", "content": sys_prompt},
{"role": "user", "content": previous_state.replace(INCORRECT_MOVE_LANGUAGE, "")},
{"role": "user", "content": previous_move},
{"role": "user", "content": current_state + valid_moves}
]
else:
# Append the new game state as a user message
messages.append({"role": "user", "content": current_state + valid_moves})
time.sleep(seconds_sleep_per_step) ## Prevent myself from hitting rate limits.
if current_state == ending_state:
print(f"Completed the game at {num_iterations} iterations")
else:
print(f"Stoped gameplay by reaching the max number of iterations ({max_num_iterations})")
One note about tokenization: I put spaces between the “disc” numbers in the stack to ensure that those numbers are broken up into different tokens, but the actual moves (e.g. “3C”) don’t need a space between them because the letters and numbers are broken into different tokens automatically.
If you’re curious about how text gets tokenized for openAI models, you can check out this cool tool that visualizes how things get tokenized.
llm_plays_tower_hanoi(num_discs = 3, model = "gpt-4.1")
You are a puzzle master completing the tower of Hanoi puzzle.
This puzzle will be represented to you in this fashion:
A | 2 1 0
B |
C |
There are three stacks: A, B, and C.
There are 3 discs on the stacks, represented as numbers, with 0 being the smallest and 2 being the largest. Each digit is a disc
Your goal is to move all of the discs from stack A to stack C, in the proper order.
You must follow the following rules:
1. Only one disc may be moved at a time.
2. Each move consists of taking the top disc from one of the stacks and placing it on the top of another stack or on an empty stack.
3. No disc may be placed on top of a disc that is smaller than it. (so, disk 1 cannot be placed on top of disk 0)
The game will be represented to you in an intermediate state, where the right-most disc represents the "top" of each stack.
At the start of the game, the puzzle looked like:
A | 2 1 0
B |
C |
To win, you must move all the discs to stack C, so the puzzle will look like:
A |
B |
C | 2 1 0
Your role is to plan ahead and make the best move based on the current state of the puzzle, so that the puzzle moves closer to the winning state. Think step by step and when you are ready, return your move as the last line of your response.
To make a move, state the number of the disk and the stack you want to move it to in the following manner:
```
1B
```
The move `1B` moves the disc 1 to the top of stack B. The move `0C` moves the disc 0 to the top of stack C.
Think step by step about your move, considering how your move will enable future moves to reach the winning state.
When you are ready, return your chosen move as the last line of your response.
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's analyze the current state:
A | 2 1 0
B |
C |
We need to move all discs to stack C.
The rules say only one disc can be moved at a time and a larger disc cannot go on a smaller disk.
The smallest disk (0) is at the top of stack A. To begin solving for 3 disks, the first step is always to move the smallest disc to the target position of the final stack, which is C. This frees the way for subsequent legal moves and follows the standard optimal strategy for Tower of Hanoi.
So, my move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's think step by step:
Current state:
A | 2 1
B |
C | 0
- Disk 0 is on C, which is our goal stack, but eventually disk 2 and 1 need to go there too.
- Next, we need to move disc 1 off stack A to allow us to free up disk 2, but stack C already has disk 0, so we can't put disc 1 on C because a larger disk cannot be placed on top of a smaller one.
- The only way to continue is to get disk 0 off C to open up space for larger disks later, and the only other place for disk 0 is B.
Available moves: `0B`, `0C`.
- `0C` would place disk 0 back onto C, essentially no change as it's already there.
- `0B` moves the topmost disk 0 to stack B, which is necessary for progress.
This also matches the standard Tower of Hanoi solution path.
My move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's analyze the current state and valid moves:
A | 2 1
B | 0
C |
Goal: Move all disks to stack C in order.
**Step-by-step reasoning:**
- Disk 0 is on B.
- Disk 1 is on top of disk 2 on A.
- C is empty.
We want disk 1 and disk 2 to eventually move to C, but first, disk 1 needs to be moved off A.
The valid moves are: [`1B`, `0A`, `0B`].
- `1B`: Move disk 1 to stack B, where disk 0 is currently on top. This is not allowed as 1 cannot go on top of 0.
- `0A`: Move disk 0 from B back to A. This would place it over disk 1 but doesn't make sense as we want the bigger disks on C.
- `0B`: Disk 0 is already on B, moving it to B does nothing.
But looking at rules, `1B` **would not be valid** if disk 0 is already on top of B. Since only valid moves are provided, it seems these are simply technically possible but not all optimal. (If following the input strictly, `1B` would attempt to put disk 1 on top of 0, which is not allowed.)
Therefore, the only move that makes sense is to move disk 0 back to A (`0A`), unblocking future moves for disk 1.
My move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's analyze the current state and valid moves:
Current state:
A | 2 1 0
B |
C |
Valid moves: [`1C`, `0A`, `0C`]
Let’s break down the options:
- `1C`: Move disk 1 to C. Not possible because disk 0 is on top, so disk 1 cannot be moved.
- `0A`: Move disk 0 to A. Disk 0 is already on A — this is redundant.
- `0C`: Move disk 0 to C. This is moving the topmost (smallest) disk to the target stack C.
We're at the initial state, so the correct first move is to move the smallest disk to the destination (C), which sets up further moves in the optimal solution.
Therefore, the best move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's analyze:
Current state:
A | 2 1
B |
C | 0
Valid moves: [`0B`, `0C`]
- Disk 0 is on C (goal stack), but to move the other disks, we will need to clear C.
- If we move `0C`, nothing changes since disk 0 is already on C.
- If we move `0B`, we move disk 0 to B, clearing C so that larger disks can be placed there later.
This move is necessary to progress according to the standard Tower of Hanoi method.
My move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's evaluate the situation:
Current state:
A | 2 1
B | 0
C |
Valid moves: [`1B`, `0A`, `0B`]
- Disk 0 is on top of B.
- Disk 1 is on top of disk 2 on A.
- We want to free disk 1 so that we can start transferring the larger disks toward C eventually.
Move-by-move analysis:
- `0A`: Moves disk 0 from B back to A, making the stacks A | 2 1 0, B | , C |. This basically returns to a previous state.
- `0B`: Disk 0 is already on B, so this is redundant.
- `1B`: Moving disk 1 to B would only be valid if disk 0 is not already on B, because you can’t place 1 on top of 0. So this is not a legal move under classic rules. But since it's listed, we'll assume for the puzzle context, maybe the moves are simply provided as "theoretically" possible and up to us to pick the valid one.
However, the only truly available action that advances the solution is to move disk 0 back to A so that B is open to receive disk 1 on another move.
Thus, the best move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let’s look at the current state:
A | 2 1 0
B |
C |
Valid moves are: [`1C`, `0A`, `0C`]
Let’s examine the moves:
- `1C`: Disk 1 is not accessible--disk 0 is on top of it, so you can't move it yet.
- `0A`: Disk 0 is already on A, so this doesn't change anything.
- `0C`: Moving disk 0 (the smallest disk) to C is the only meaningful progress for the first step in solving Tower of Hanoi.
This matches the classic first move in the puzzle and is necessary to proceed.
So, the best move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's reason carefully:
Current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
- Disk 0 (the smallest) is on C (the target peg).
- Disks 1 and 2 are still on A, but we can't move them to C until C is clear.
- The only way to clear C is to move disk 0 to another stack.
Move choices:
- `0B`: Move disk 0 from C to B. This is necessary to allow the larger disks to eventually move to C.
- `0C`: Simply puts disk 0 back onto C, changing nothing.
Therefore, the optimal move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's analyze the situation:
Current state:
A | 2 1
B | 0
C |
Valid moves: [`1B`, `0A`, `0B`]
Disc positions:
- Disk 0 is on B.
- Disk 1 is on top of 2 on A.
We want to eventually move all disks to C.
Let's consider each move:
- `1B`: This would attempt to move disk 1 from A to B, but disk 0 (smaller) is already on B. Rules prohibit placing a larger disk on top of a smaller one. So, this is not a legal move.
- `0A`: Move disk 0 from B back to A. This would make stack A: 2 1 0, B: empty, and C: empty — returning to the previous state.
- `0B`: Disk 0 is already on B; moving it to B changes nothing.
The only move that will eventually let us progress (by unblocking B so we can move disk 1 there on the next turn) is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Given the current state:
```
A | 2 1 0
B |
C |
```
Valid Moves: [`1C`, `0A`, `0C`]
Analysis:
- `1C`: Can't move disk 1; disk 0 is on top of it, according to the rules.
- `0A`: Disk 0 is already on A; moving it to A doesn’t change the puzzle.
- `0C`: Moving the top disk (smallest, disk 0) from A to C is the classic first move In Tower of Hanoi (3 disk version).
This move sets up the necessary sequence of steps to transfer the disks to C efficiently.
**The correct move is:**
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's consider the situation:
Current state:
A | 2 1
B |
C | 0
Valid moves: [`0B`, `0C`]
- Disk 0 is currently on C, which will eventually hold all disks, but right now, we need to move disk 1 and disk 2 from A to C.
- This can't be done until C is clear of smaller disks.
If we:
- `0B`: Move disk 0 from C to B, freeing up stack C for larger disks.
- `0C`: Disk 0 is already on C, so this move does nothing.
To enable progress and follow the strategy for solving the puzzle, we should move disk 0 to B.
My move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let’s break down the current situation:
A | 2 1
B | 0
C |
Valid moves: [`1B`, `0A`, `0B`]
- Disk 1 is on A, but disk 0 is on top of B, so you can't move 1 to B since it can't be placed on a smaller disk.
- `0B` is a null move since 0 is already on B.
- `0A` would move disk 0 from B back to A, making A | 2 1 0 and resetting to an earlier state.
None of these moves directly progresses the puzzle except `0A`, which is the only proper move given that `1B` is not allowed by the rules, and `0B` does nothing.
So, the best move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's carefully analyze the current state and options:
A | 2 1 0
B |
C |
Valid moves: [`1C`, `0A`, `0C`]
- `1C`: Disk 1 is not at the top of stack A (disk 0 is), so this shouldn't be permitted yet.
- `0A`: Disk 0 is already on A, so moving it to A changes nothing.
- `0C`: Move disk 0 to C, which is both allowed and the classic first step in solving the 3-disk Tower of Hanoi puzzle.
This move sets the stage for progressing toward the goal.
Thus, the best move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's review the state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
Disk 0 (smallest) is currently on C (the target stack), but disks 1 and 2 must move from A to C, so we need to clear C to do this.
- `0C`: Moves disk 0 from C to C, which does nothing.
- `0B`: Moves disk 0 from C to B, freeing C for the larger disks.
This is the standard next step in Tower of Hanoi with 3 disks.
So, the correct move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's think step by step:
**Current State:**
```
A | 2 1
B | 0
C |
```
Discs: 0 on B, 1 and 2 on A.
**Valid Moves:**
- `1B`: Move disk 1 to B (NOT allowed, because disk 0 is already there and you cannot place a larger disk on a smaller disk).
- `0A`: Move disk 0 from B back to A. This resets to an earlier position (A | 2 1 0, B | , C |).
- `0B`: Disk 0 is already on B (no change).
**Analysis:**
The only move that is legal and creates a new state is to move disk 0 from B back to A. While this repeats a previous state, it's the only way to proceed and try to reach a different path.
So the move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let’s look at the current state:
A | 2 1 0
B |
C |
Valid Moves: [`1C`, `0A`, `0C`]
- `1C`: Can't move disk 1 yet—disk 0 is on top.
- `0A`: Disk 0 is already on A, so this move has no effect.
- `0C`: Moving disk 0 from A to C is the optimal first move in the Tower of Hanoi.
Therefore, the best move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's analyze the situation step by step:
Current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
Disk 0 is on C, but in order to move disk 1 or disk 2 to C, C must be clear (since you can't put a larger disk on a smaller one).
- Moving `0C` would do nothing, as disk 0 is already on C.
- Moving `0B` will move disk 0 from C to B, clearing C and allowing the larger disks to be moved there in upcoming steps.
**Therefore, the best move is:**
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let’s analyze the state and possible moves:
**Current state:**
```
A | 2 1
B | 0
C |
```
**Valid Moves:** [`1B`, `0A`, `0B`]
Disk 1 is on stack A, but you cannot put it on stack B because disk 0 (smaller) is already there—this is not a legal move.
`0B` is pointless (disk 0 is already on B).
The only move that changes the state is `0A`, which moves disk 0 from B back to A:
- After `0A`:
A | 2 1 0
B |
C |
Although this returns to a previous state, it's the only move that adheres to the rules and progresses the puzzle given current options.
**So, my move is:**
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's review the state and moves:
Current state:
A | 2 1 0
B |
C |
Valid moves:
- `1C`: Not possible; disk 1 is not on top.
- `0A`: Disk 0 is already on A; move does nothing.
- `0C`: Move disk 0 from A to C, which is the correct first step in the Tower of Hanoi puzzle.
The optimal move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's examine the state and valid moves:
Current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
- Disk 0 is on C (target stack).
- Ultimately, we want to move disks 1 and 2 to C as well, but can’t do so until C is clear of smaller disks.
- Moving `0C` would mean putting disk 0 back onto C, but it's already there, so that changes nothing.
- Moving `0B` will move disk 0 off C and onto B, which is necessary to free up C for the larger disks.
Therefore, in order to make progress toward the goal:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's break down the situation:
Current state:
A | 2 1
B | 0
C |
Valid Moves: [`1B`, `0A`, `0B`]
- `1B`: Move disk 1 from A to B, but disk 0 is already on top of B. By the rules, you cannot place a larger disk (1) on top of a smaller disk (0), so this is not a legal move.
- `0A`: Move disk 0 from B back to A. This returns disk 0 to A, recreating the initial stack.
- `0B`: Disk 0 is already on B, so this move does nothing.
Of the valid moves listed, only `0A` is meaningful and adheres to the rules.
Therefore, the best move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's analyze the current state and options:
Current state:
```
A | 2 1 0
B |
C |
```
Valid Moves: [`1C`, `0A`, `0C`]
- `1C`: Disk 1 is not on top (disk 0 is), so this is not a valid move now.
- `0A`: Disk 0 is already on A; this changes nothing.
- `0C`: Move disk 0 from A to C, which is the classic first move in the Tower of Hanoi for 3 disks and sets up all future moves.
Therefore, the optimal move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's analyze the current state and valid moves:
Current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
- Disk 0 is currently on C.
- To progress, we need to move disks 1 and 2 to C, but they can't be placed on C until it's empty.
Considering the available moves:
- `0C`: Moving disk 0 to C does nothing, since it's already there.
- `0B`: Moving disk 0 to B clears C for future moves, enabling us to eventually move disks 1 and 2 to C.
Therefore, the best move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's evaluate the current situation:
Current state:
A | 2 1
B | 0
C |
Valid Moves: [`1B`, `0A`, `0B`]
Analysis:
- `1B`: Move disk 1 to B, but disk 0 is already on B, and a larger disk can't be placed on a smaller one. Invalid by Tower of Hanoi rules.
- `0A`: Move disk 0 from B back to A. This reverses the previous move and restores the starting stack.
- `0B`: Disk 0 is already on B, so this changes nothing.
Of the options, `0A` is the only rule-following move available.
**My move is:**
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's analyze the state and moves:
Current state:
A | 2 1 0
B |
C |
Valid Moves:
- `1C`: Disk 1 is not accessible (disk 0 is on top).
- `0A`: Disk 0 is already on stack A, so this is redundant.
- `0C`: Move the top (smallest) disk 0 from A to C, which is the standard opening move in the Tower of Hanoi puzzle.
Making `0C` sets us up for transferring the rest of the disks efficiently.
**The optimal move is:**
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's look at the current state and valid moves:
Current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
- Disk 0 is on C.
- To eventually move disks 1 and 2 to C, C must be clear of any smaller disk.
So:
- `0B`: Move disk 0 from C to B, freeing up C for the larger disks.
- `0C`: This does nothing, since disk 0 is already on C.
To enable progress toward the solution, the correct move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's analyze the situation:
Current state:
A | 2 1
B | 0
C |
Valid Moves: [`1B`, `0A`, `0B`]
- `1B`: Attempting to move disk 1 to B is **not allowed** because disk 0 (smaller) is already on B.
- `0A`: Move disk 0 from B back to A. This would put A | 2 1 0, B | , C |.
- `0B`: Disk 0 is already on B (no change).
The only move that follows the rules and changes the state is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's analyze the current state and the available moves:
Current state:
A | 2 1 0
B |
C |
Valid Moves: [`1C`, `0A`, `0C`]
Disk 0 is at the top of stack A.
- `1C`: Disk 1 can't be moved yet because it's under disk 0.
- `0A`: Disk 0 is already on stack A; this changes nothing.
- `0C`: Move disk 0 from A to C, which is the classic and optimal first move in a 3-disk Tower of Hanoi.
**Therefore, the best move is:**
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's analyze:
Current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
- Disk 0 is currently on C.
- To continue the Tower of Hanoi solution, we need to eventually move disks 1 and 2 to C, but we can only do so when C is clear.
So:
- `0B`: Moves disk 0 from C to B, clearing C for the larger disks.
- `0C`: Disk 0 remains on C, which doesn't help progress.
Therefore, the best move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's analyze the current state and the valid moves:
Current state:
A | 2 1
B | 0
C |
Valid Moves: [`1B`, `0A`, `0B`]
- `1B`: Move disk 1 from A to B. However, B already has disk 0 (smaller) on top, and you can't put a larger disk on a smaller one. This is not allowed by the rules.
- `0A`: Move disk 0 from B back to A. This would restore the initial full stack on A. It's not progress, but with the current state, it's the only legal move that changes the configuration.
- `0B`: Disk 0 is already on B, so this doesn't change anything.
**Therefore, the only legal move here is:**
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's carefully consider the state and options:
Current state:
```
A | 2 1 0
B |
C |
```
Valid Moves: [`1C`, `0A`, `0C`]
- `1C`: Disc 1 is not accessible because disc 0 is on top.
- `0A`: Disc 0 is already on A — this move changes nothing.
- `0C`: Move disc 0 from A to C. This is the optimal and classic first move for a 3-disc Tower of Hanoi puzzle.
Therefore, the best move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's analyze the situation:
Current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
- Disk 0 is on C, but C needs to be empty to move larger disks from A onto it.
- Moving `0C` keeps disk 0 on C, no progress.
- Moving `0B` places disk 0 on B and clears C so we can move the larger disks to C in the future.
The move that allows us to progress is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's analyze the situation step by step:
Current state:
A | 2 1
B | 0
C |
Valid Moves: [`1B`, `0A`, `0B`]
- `1B`: Trying to move disk 1 to B is not allowed, since disk 0 (smaller) is already on B.
- `0A`: Move disk 0 from B back to A. This would revert to a previously seen state (A | 2 1 0, B | , C |), but it's the only actionable and legal move available.
- `0B`: Disk 0 is already on B; this does not change anything.
Therefore, the only way to continue is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's analyze the current state and your options:
A | 2 1 0
B |
C |
Valid Moves: [`1C`, `0A`, `0C`]
- `1C`: Cannot be played because disk 0 is on top of disk 1.
- `0A`: Would move disk 0 from A to A, which changes nothing.
- `0C`: Moves the smallest disk (0) from A to C. This is the essential first move in the Tower of Hanoi and enables all subsequent progress.
Therefore, the best move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's look at the state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
- Disk 0 is currently on C.
- To progress, we need to move disks 1 and 2 to C, but cannot do so until C is emptied.
Therefore, the optimal move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's analyze the state and possible moves:
Current state:
A | 2 1
B | 0
C |
Valid Moves: [`1B`, `0A`, `0B`]
- `1B`: Cannot move disk 1 to B because disk 0 is already there (no larger disk on top of a smaller one).
- `0A`: Move disk 0 from B back to A. This is the only legal move that changes the configuration, though it resets to the previous state.
- `0B`: Disk 0 is already on B—no change.
Therefore, the only rule-valid move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's analyze the situation:
Current state:
A | 2 1 0
B |
C |
Valid Moves: [`1C`, `0A`, `0C`]
- `1C`: Disk 1 cannot be moved (disk 0 is on top).
- `0A`: Disk 0 is already on A; no effect.
- `0C`: Move disk 0 from A to C — this is the classic first move and opens the path for future progress.
So, the best and only meaningful move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's consider the current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
- Disk 0 is on C. To move the larger disks (1 and 2) to C, we must first clear C.
- `0C` would simply leave disk 0 on C (no change).
- `0B` would move disk 0 to B, opening up C for the next necessary moves.
To make progress, the correct move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's analyze the current state and the possible moves:
Current state:
A | 2 1
B | 0
C |
Valid Moves: [`1B`, `0A`, `0B`]
- `1B`: Not valid—disk 1 cannot go on disk 0 (a larger disk cannot be placed on a smaller disk).
- `0A`: Move disk 0 from B back to A. This puts us back to the original state, but with the current restrictions, it allows for different options in following moves.
- `0B`: Disk 0 is already on B, so this move changes nothing.
Therefore, the only legal and state-changing move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's consider the current state and the allowed moves:
A | 2 1 0
B |
C |
Valid Moves: [`1C`, `0A`, `0C`]
- `1C`: Not possible, as disk 1 is not on top of A (disk 0 is above it).
- `0A`: Disk 0 is already on A; this move changes nothing.
- `0C`: Move disk 0 from A to C — this is the optimal first move in the Tower of Hanoi for 3 disks.
**The correct move is:**
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's review the state and available moves:
Current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
Disk 0 is already on C. To proceed, we must eventually move disks 1 and 2 to C, but can't do so until C is clear.
- `0C` (disk 0 to C) would have no effect.
- `0B` (disk 0 to B) clears stack C, allowing us to progress with moving disks 1 and 2 to C.
Therefore, the best move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's analyze the current situation:
Current state:
A | 2 1
B | 0
C |
Valid Moves: [`1B`, `0A`, `0B`]
- `1B`: Not allowed — disk 1 (larger) cannot go on disk 0 (smaller).
- `0A`: Move disk 0 from B back to A. This would revert to a previous state, but it's the only rule-abiding move that changes the configuration.
- `0B`: Disk 0 is already on B, so this doesn't change anything.
Therefore, the only valid move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's look at the current state:
A | 2 1 0
B |
C |
Valid Moves: [`1C`, `0A`, `0C`]
- `1C`: Not possible, as disk 0 (the smallest) is on top of disk 1 on stack A.
- `0A`: Disk 0 is already on A; this move changes nothing.
- `0C`: Move disk 0 from A to C (the classic and optimal first move in 3-disk Tower of Hanoi).
Therefore, the best move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's analyze the current state:
A | 2 1
B |
C | 0
Valid Moves: [`0B`, `0C`]
- Disk 0 is on C (our goal stack), but we need to move disks 1 and 2 there in order.
- To do this, C must be empty before we can move a larger disk onto it.
- `0B`: Moves disk 0 from C to B, clearing C for future necessary moves.
- `0C`: Keeps disk 0 on C and makes no progress.
Therefore, the correct move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's look at the current state and choices:
A | 2 1
B | 0
C |
Valid Moves: [`1B`, `0A`, `0B`]
- `1B`: Not allowed, because disk 1 (larger) can't go on disk 0 (smaller).
- `0A`: Move disk 0 from B back to A. This is the only move that changes the configuration.
- `0B`: Disk 0 is already on B; move has no effect.
Therefore, the only valid move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's analyze the given state:
A | 2 1 0
B |
C |
Valid Moves: [`1C`, `0A`, `0C`]
- `1C` is not possible since disk 1 is not on the top of A (disk 0 is on top).
- `0A` simply puts disk 0 back on A, where it already is. This doesn't change the state.
- `0C` is the classic first move: move disk 0 (the smallest, currently on top) from A to C.
Therefore, the optimal move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's examine the current state and the available moves:
```
A | 2 1
B |
C | 0
```
Valid Moves: [`0B`, `0C`]
- Disk 0 is on C. To move the larger disks (1 and 2) onto C, we need to free C of disk 0.
- Moving `0C` would keep disk 0 on C (no progress).
- Moving `0B` will remove disk 0 from C and put it on B, opening C for the larger disks.
Therefore, the best move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2 1
B | 0
C |
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let's analyze the current state and available moves:
Current state:
A | 2 1
B | 0
C |
Valid moves: [`1B`, `0A`, `0B`]
Disk 1 is on A, but cannot be moved to B, because disk 0 (which is smaller) is already on top of B. You cannot place a larger disk on a smaller one.
- `1B`: Not permitted (would break the rules).
- `0A`: Move disk 0 from B back to A (undoing the previous move, but the only available legal move that changes the puzzle state).
- `0B`: Disk 0 is already on B (no change).
Therefore, the best move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's analyze the current state and the available moves:
A | 2 1 0
B |
C |
Valid Moves: [`1C`, `0A`, `0C`]
- `1C`: Disk 1 is not on top of stack A; disk 0 is blocking it, so this isn't currently possible.
- `0A`: Disk 0 is already on stack A; moving it to A changes nothing.
- `0C`: Moving disk 0 (the smallest, currently on top of A) to C is the opening move in the optimal Tower of Hanoi strategy.
Therefore, the best move is:
```
0C
```
Stoped gameplay by reaching the max number of iterations (50)
The model frequently made moves that would not possibly help move to a winning state. In fact, the model only moved disc 0 the whole game!
The model does follow the directions to “think step by step”, but it is not capable of making correct logical deductions as part of that process.
llm_plays_tower_hanoi(num_discs = 3, model = "gpt-4.1", restrict_token_length = True)
You are a puzzle master completing the tower of Hanoi puzzle.
This puzzle will be represented to you in this fashion:
A | 2 1 0
B |
C |
There are three stacks: A, B, and C.
There are 3 discs on the stacks, represented as numbers, with 0 being the smallest and 2 being the largest. Each digit is a disc
Your goal is to move all of the discs from stack A to stack C, in the proper order.
You must follow the following rules:
1. Only one disc may be moved at a time.
2. Each move consists of taking the top disc from one of the stacks and placing it on the top of another stack or on an empty stack.
3. No disc may be placed on top of a disc that is smaller than it. (so, disk 1 cannot be placed on top of disk 0)
The game will be represented to you in an intermediate state, where the right-most disc represents the "top" of each stack.
At the start of the game, the puzzle looked like:
A | 2 1 0
B |
C |
To win, you must move all the discs to stack C, so the puzzle will look like:
A |
B |
C | 2 1 0
Your role is to plan ahead and make the best move based on the current state of the puzzle, so that the puzzle moves closer to the winning state. Think step by step and when you are ready, return your move as the last line of your response.
To make a move, state the number of the disk and the stack you want to move it to in the following manner:
```
1B
```
The move `1B` moves the disc 1 to the top of stack B. The move `0C` moves the disc 0 to the top of stack C.
Think step by step about your move, considering how your move will enable future moves to reach the winning state.
When you are ready, return your chosen move as the last line of your response.
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
Let's think step by step:
1. The goal is to move all discs from A to C.
2. The rules allow only moving the top disc of each stack (the right-most digit in the listed stack). In this case, the top of stack A is 0.
3. To get all discs to C, we should follow the classic Tower of Hanoi algorithm for 3 discs. The first step is always to move the smallest disc (disc 0) to the target stack C. This opens up the way for moving the other discs.
So:
- Current top of A: 0 (can be moved)
- B and C are empty.
Moving disc 0 to stack C is a correct first step.
My move is:
```
0C
```
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let's analyze the state step by step:
1. Stack A has discs 2 and 1, with 1 on top (since 1 is the right-most).
2. Stack C only has disc 0, which is now blocking the use of C for the larger discs until disc 0 is moved.
3. The possible valid moves are moving the top-most disc from A (which is disc 1) or from C (which is disc 0).
Now, to follow the optimal sequence of the Tower of Hanoi for 3 discs, after moving disc 0 to C, we should move the next smallest disc (disc 1) to stack B (the auxiliary stack). This is because the target peg C is already occupied by the smallest disc, and we can't put disc 1 on top of 0 (it's not allowed to place a larger disc on a smaller one).
So, the best move is to move disc 1 from stack A to stack B.
My move is:
```
1B
```
--------------------
The current state of the game is:
```
A | 2
B | 1
C | 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
Let's break down the current situation:
- Stack A has only the largest disc (2).
- Stack B has disc 1.
- Stack C has disc 0.
The valid moves are: move 1 to B (but it's already there, so not helpful), or move the smallest disc, 0, either to A or B.
Let’s analyze:
- If we want to eventually move disc 2 to C, both discs 0 and 1 need to be moved away from C and B so 2 can travel to C.
- In the standard Tower of Hanoi solution:
1. Move 0 from C to B to free up C.
2. Move 2 from A to C.
3. Finish stacking 1 and 0 from B to C.
To proceed, I need to start by moving 0 from C to B, stacking it on top of 1. This is a legal move and is the correct next step in the standard solution.
My move is:
```
0B
```
--------------------
The current state of the game is:
```
A | 2
B | 1 0
C |
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
Let's consider the new state:
- Stack A: 2
- Stack B: 1 (bottom), 0 (top)
- Stack C: empty
Valid moves: Move 1 to A, 0 to A, 0 to B (but 0 is already on top of B, so that's not useful).
Here's the goal: get all discs to C, following the rules.
Next, we need to move disc 2 from A to C eventually, but this can only happen when C is empty (which it is now), and neither disc 0 nor 1 is on C. But currently, we can't move 2 yet, because it's not an available move this turn (only the top disc can be moved, and 2 is the only one on A, so actually, 2C is a valid move, but you only listed 1A, 0A, 0B as valid moves—perhaps by mistake).
Given your list of valid moves, let's consider the optimal choice:
- Moving 0A: Moves 0 from B to A, placing it on top of 2. That's legal since 0 < 2.
- Moving 1A: Only possible if 1 is the top of its stack, but 1 is under 0.
- 0B: Disc 0 is already on B.
So only 0A is possible.
Strategically, moving disc 0 to A gets B ready for disc 2 to land on C, which matches the correct sequence in the solution.
The best move is:
```
0A
```
--------------------
The current state of the game is:
```
A | 2 0
B | 1
C |
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
Let's break down the options:
- A: 2 (bottom), 0 (top)
- B: 1 (top)
- C: empty
Valid moves: `2C` (move disc 2 to C), `0A` (move disc 0 to A, but it's already on A), `0C` (move disc 0 to C).
Let's consider each:
1. **2C:**
Move disc 2 from A to C.
However, disc 0 is currently on top of disc 2. Only the top disc (0) on stack A can be moved, so this move is not legal at this moment unless the representation is "rightmost is top". If "2 0" on A means disc 2 is bottom, disc 0 is top, then only 0 may be moved from A.
2. **0A:**
Moving disc 0 to A, but it's already on top of A. This does nothing.
3. **0C:**
Move disc 0 from A to C, results in:
- A: 2
- B: 1
- C: 0
After this, either disc 2 or 1 can move depending on next moves.
**Strategic Analysis:**
Moving 0 to C is a standard step in the Tower of Hanoi 3-disc solution at this stage. Once 0 is on C, 2 can eventually move to C after dealing with disc 1.
**Best Move:**
Move disc 0 to C.
```
0C
```
--------------------
The current state of the game is:
```
A | 2
B | 1
C | 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
Let's analyze the current state:
- **A:** 2 (top)
- **B:** 1 (top)
- **C:** 0 (top)
Remember, "right-most is top," so C's top disc is 0.
Valid moves:
- `0B`: Move disc 0 from C to B (since 0 is smaller than 1, this is valid)
- `0C`: Move disc 0 from C to C (disc 0 is already at C, so this does nothing)
- `1C`: Move disc 1 from B to C (but disc 1 can't go on top of disc 0, since 1 > 0 and smaller must be on top, so this move is invalid or not achievable under the rules)
Let's reason through the valid ones:
1. **`0B`**:
Move disc 0 from C to B (on top of 1).
New state:
- A: 2
- B: 1 0 (0 on top)
- C: (empty)
After this, only top disc 0 on B, 2 on A can move.
2. **`1C`**:
Attempting to move disc 1 from B to C, but since 0 is currently on C, and you cannot place 1 on 0 (must be smaller disc on top), so this move is not allowed per rules.
**Strategic analysis:**
Moving 0 back to B may seem like a regression, but it is in fact a necessary step in the sequence to free up the largest disc (2) on A.
The standard 3-disc Tower of Hanoi moves the smallest disc back and forth to set up the larger disc’s move.
**Best Move:**
```
0B
```
--------------------
The current state of the game is:
```
A | 2
B | 1 0
C |
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
Let's analyze the current state and valid moves:
- **A:** 2
- **B:** 1 0 (0 is the top)
- **C:** (empty)
Valid moves:
- `1A`: Move 1 from B to A (would put 1 on top of 2; valid since 1 < 2)
- `0A`: Move 0 from B to A (would put 0 on top of 2; valid since 0 < 2)
- `0B`: Move 0 from B to B (no effect, so not useful)
**Goal:** Eventually move all discs to C in order 2 1 0.
Let’s reason step by step:
- If we do `0A`, A will be 2 0. B will be 1.
Then what?
- We can’t move 1 to A (would place on 0 — not allowed).
- We could move 1 to C, but then we still have to sort out 2 and 0.
- If we do `1A`, A will be 2 1. B will be 0.
- Now, 0 can only move to C (empty), freeing space.
- Continuing this way will allow us to finish the puzzle.
So, moving 1 to A sets up moving 0 directly to C and getting the largest disc to C at the end.
**Best move:**
```
1A
```
--------------------
Invalid move. The current state of the game is:
```
A | 2
B | 1 0
C |
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
Let's look carefully at the current state and valid moves:
```
A | 2
B | 1 0
C |
```
- Stack A: 2 (2 is at the top)
- Stack B: 1 0 (0 is at the top)
- Stack C: (empty)
**Valid Moves:**
- `2C`: Move disk 2 from A to C
- `0A`: Move disk 0 from B to A
- `0C`: Move disk 0 from B to C
**Immediate goal:** Move all discs to C.
Let's consider each move:
### Move `2C`
Move disk 2 from A to C
Resulting state:
```
A |
B | 1 0
C | 2
```
Now, B's top is 0, and we can move it to either A or C. This move helps get the largest disk to its destination stack.
### Move `0A`
Move 0 from B to A
Resulting state:
```
A | 2 0
B | 1
C |
```
Now, A has 2 on the bottom and 0 on top. Moving 1 next is blocked since 1 cannot go on top of 0.
### Move `0C`
Move 0 from B to C
Resulting state:
```
A | 2
B | 1
C | 0
```
Now, 2 can go to C (since 2 > 0 not allowed), but otherwise 2 would have to wait until 1 is moved.
### Conclusion
The optimal move is `2C`. This gets the largest disk to its target stack C, which is a key step in solving the puzzle.
**My move:**
```
2C
```
--------------------
The current state of the game is:
```
A |
B | 1 0
C | 2
```
Valid Moves are: [`0A`, `0C`, `2A`]
--------------------
Let’s analyze the current state and the valid moves:
```
A |
B | 1 0
C | 2
```
- A is empty.
- B has 1 (bottom) and 0 (top).
- C has 2.
**Valid Moves:** `0A`, `0C`
Looking closer:
- `2C` is not valid since disk 2 is already on C.
- The only two moves are `0A` (move disk 0 to A) and `0C` (move disk 0 to C).
### Goal
To eventually stack 2, 1, and 0 (from bottom to top) on stack C.
### Consider each move:
#### Move `0A`
Move disk 0 from B to A:
Result:
```
A | 0
B | 1
C | 2
```
Now, the only allowed move is likely to move 1 somewhere (depending on the rules), but 1 can be moved to either A or C. But you want 1 to go to C eventually, on top of 2 (since that's the end-goal order).
#### Move `0C`
Move disk 0 from B to C:
Result:
```
A |
B | 1
C | 2 0
```
Now, the order on C is 2 (bottom), 0 (top). But in the end, you want 2-1-0 order, so you still need 1 above 2 and below 0, which isn't possible directly since you cannot place 1 on top of 0.
### Optimal Move
Moving `0A` sets you up for the correct stacking order and leaves C open for 1 to be moved to it next, then 0 can eventually go on top.
**Best Move:**
```
0A
```
--------------------
The current state of the game is:
```
A | 0
B | 1
C | 2
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
Let's analyze the situation:
```
A | 0
B | 1
C | 2
```
- Stack A: 0 (smallest)
- Stack B: 1 (middle)
- Stack C: 2 (largest)
**Valid Moves:** `0A` (keep 0 on A, ineffective), `0C` (move 0 to C), `2A` (move 2 to A).
## Step by Step Reasoning
Our goal: all discs on C in order 2 1 0 (bottom to top).
### Review Moves
1. **Move `0A`**
- Disc 0 is already on A; leaving it there makes no progress.
2. **Move `0C`**
- Move 0 from A to C.
- Result:
```
A |
B | 1
C | 2 0
```
- Now on C: [2 (bottom), 0 (top)].
- Next, we need disc 1 on C, but it's not possible to put 1 on top of 0 (since 1 > 0). This may temporarily block future moves.
3. **Move `2A`**
- Move 2 from C to A.
- Result:
```
A | 2 0
B | 1
C |
```
- Now A: [2 (bottom), 0 (top)].
- B: 1.
- You cannot place 1 on A (can only go on empty stacks or on 2), but to transfer all to C, this is moving backward.
## Optimal Path Analysis
The primary objective is to get 1 on top of 2 in C, followed by 0. But our current stack order means before moving 0 to C, we need 1 to go to C first.
### Think Ahead
If we move 2 from C to A:
```
A | 2 0
B | 1
C |
```
Next, we could potentially move 1 to C, then 0 somewhere, and then 2 to C, building up to the correct arrangement.
#### Let's Check that Sequence:
After `2A`:
```
A | 2 0
B | 1
C |
```
Valid moves would be: move 1 to C.
Move 1 to C:
```
A | 2 0
B |
C | 1
```
Now, 0 can be moved from A to B:
```
A | 2
B | 0
C | 1
```
Now, 2 can be moved from A to C:
```
A |
B | 0
C | 1 2
```
Then, move 0 from B to C:
```
A |
B |
C | 1 2 0
```
This would result in an incorrect order on C: [1, 2, 0], which is not winning (should be [2, 1, 0]), but currently, with the moves available, moving 2A allows progress toward combining the discs in proper order.
### Hence, the best path to progress is:
```
2A
```
--------------------
Invalid move. The current state of the game is:
```
A | 0
B | 1
C | 2
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
Let's reconsider each valid move based on the rules and our objective:
```
A | 0
B | 1
C | 2
```
**Valid Moves:** [`0B`, `0C`, `1C`]
- We want all discs in order 2 1 0 on C.
**Analysis:**
- `0B`: Moves disc 0 to B, producing:
```
A |
B | 1 0
C | 2
```
We still need to eventually get 1 and 0 onto C, but now 1 and 0 are stacked together, which doesn’t immediately enable placing 2 or 1 onto C.
- `0C`: Moves disc 0 to C, producing:
```
A |
B | 1
C | 2 0
```
Now C has 2 at the bottom, 0 on top. Can't place 1 on C (since you can't place 1 on top of 0). This also doesn’t bring us closer to stacking them in order.
- `1C`: Moves disc 1 to C, producing:
```
A | 0
B |
C | 2 1
```
This stacks 1 on top of 2 on C, getting them into the correct order (since 0 must go on top last).
From here, we could then finish by moving 0 from A to C:
```
A |
B |
C | 2 1 0
```
Which is the goal state!
Therefore, the optimal move is:
```
1C
```
--------------------
The current state of the game is:
```
A | 0
B |
C | 2 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
Let's carefully analyze the situation and our options:
```
A | 0
B |
C | 2 1
```
Our goal is:
```
A |
B |
C | 2 1 0
```
**Valid Moves:** [`0B`, `0C`, `1C`]
- `0B`: Moves the 0 disk to stack B:
```
A |
B | 0
C | 2 1
```
This temporarily gets 0 out of the way, but 1 is already on top of 2 on C, so this doesn't advance us toward the goal.
- `0C`: Moves the 0 disk to stack C:
```
A |
B |
C | 2 1 0
```
This is exactly the goal state! All discs are stacked on C in the correct order.
- `1C`: Not possible, since disk 1 is already on C.
**Conclusion:**
The best move is to move disk 0 to stack C.
```
0C
```
Completed the game at 12 iterations
Here, the model successfully completed the game in 12 iterations (only 5 more moves than the 7 minimum needed to complete the game with 3 moves. However, there were moves where there was a clear “correct” move (disc 2 was at the top of one stack and stack B was empty) and the model did not take the move.
I unfortunately only have access to o4-mini (my openAI developer account isn’t connected to an organization) so this is the highest-quality LRM I have access to.
llm_plays_tower_hanoi(num_discs = 3, model = "o4-mini")
You are a puzzle master completing the tower of Hanoi puzzle.
This puzzle will be represented to you in this fashion:
A | 2 1 0
B |
C |
There are three stacks: A, B, and C.
There are 3 discs on the stacks, represented as numbers, with 0 being the smallest and 2 being the largest. Each digit is a disc
Your goal is to move all of the discs from stack A to stack C, in the proper order.
You must follow the following rules:
1. Only one disc may be moved at a time.
2. Each move consists of taking the top disc from one of the stacks and placing it on the top of another stack or on an empty stack.
3. No disc may be placed on top of a disc that is smaller than it. (so, disk 1 cannot be placed on top of disk 0)
The game will be represented to you in an intermediate state, where the right-most disc represents the "top" of each stack.
At the start of the game, the puzzle looked like:
A | 2 1 0
B |
C |
To win, you must move all the discs to stack C, so the puzzle will look like:
A |
B |
C | 2 1 0
Your role is to plan ahead and make the best move based on the current state of the puzzle, so that the puzzle moves closer to the winning state. Think step by step and when you are ready, return your move as the last line of your response.
To make a move, state the number of the disk and the stack you want to move it to in the following manner:
```
1B
```
The move `1B` moves the disc 1 to the top of stack B. The move `0C` moves the disc 0 to the top of stack C.
Think step by step about your move, considering how your move will enable future moves to reach the winning state.
When you are ready, return your chosen move as the last line of your response.
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
We need to free up the largest disc (2) so it can eventually move to C, but we can only move one disc at a time and never place a larger on a smaller. The optimal sequence for three disks starts by moving the smallest disk to the destination peg (C), since there’s an odd number of disks. This sets up the alternating pattern of moves needed.
Thus, the first move is:
0C
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
We now have A: (2,1), B: (), C: (0). Our ultimate goal remains to shift the entire stack from A over to C in proper order. With three disks, the minimal-move solution after moving the smallest disk (0) to C is:
1. Move disk 1 from A to B
2. Move disk 0 from C to B
3. Move disk 2 from A to C
4. Move disk 0 from B to A
5. Move disk 1 from B to C
6. Move disk 0 from A to C
We’ve completed step 1 (0→C). The very next move (step 2 in the full sequence) is to relocate disk 1 off of A so disk 2 can later move freely. The only legal spot for disk 1 is peg B.
Therefore, the best next move is:
1B
--------------------
The current state of the game is:
```
A | 2
B | 1
C | 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
We now have A: (2), B: (1), C: (0). To free peg C for the largest disk and follow the optimal 3‐disk sequence, we must move disk 0 off of C onto B (on top of disk 1). This sets up the next move of disk 2 to C.
0B
--------------------
The current state of the game is:
```
A | 2
B | 1 0
C |
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
We’ve just moved the “tower” of two smaller discs (1,0) onto B so that 2 could go to C—but because 2→C isn’t offered right now, we continue the next sub‐solution: move the 2‐disc stack from B to C using A as spare.
For a 2‐disc transfer from B→C, the first step is to move the smallest disc (0) off B onto A.
Therefore, the best move is:
0A
--------------------
The current state of the game is:
```
A | 2 0
B | 1
C |
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We need to free disk 2 on A so it can move to C. That means first lifting the small disk 0 off of A. Moving 0 onto C (which is empty) both clears A and puts 0 in its final destination peg.
Therefore, the best move is:
0C
--------------------
The current state of the game is:
```
A | 2
B | 1
C | 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
We must free up peg C so that we can later move the largest disc (2) there. The only way to clear the smallest disc (0) off of C is to move it onto B (where it can sit on 1).
Thus the correct move is:
0B
--------------------
The current state of the game is:
```
A | 2
B | 1 0
C |
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
To move the two‐disk stack from B→C using A as auxiliary, the first step is to free disk 0 by moving it from B to A.
0A
--------------------
The current state of the game is:
```
A | 2 0
B | 1
C |
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We now have the largest disk (2) sitting unobstructed on A and C is empty. Our plan is to place disk 2 on C in its final position, then transfer the two smaller disks atop it. The only move that advances us toward that is:
2C
--------------------
Invalid move. The current state of the game is:
```
A | 2 0
B | 1
C |
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We need to free disk 2 so it can go to C, and at the same time stack disks 1 and 0 together on B so they’re ready to drop on top of 2 later. The only way to do that is to move the smallest disc off A onto B:
0B
--------------------
The current state of the game is:
```
A | 2
B | 1 0
C |
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
We still can’t touch the largest disc until both smaller ones are out of the way. The only legal move right now is to take disk 0 (top of B) and move it onto the empty peg C. This clears disk 1 so it can later move—and ultimately frees disk 2 for its move to C.
0C
--------------------
The current state of the game is:
```
A | 2
B | 1
C | 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
We need to clear the smallest disc off C so we can eventually move the largest disc. Among the legal options, only moving disk 0 from C to A actually changes the position.
Thus, the move is:
0A
--------------------
The current state of the game is:
```
A | 2 0
B | 1
C |
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We need to free disk 0 from peg A so we can eventually move disk 2. The only legal, non-trivial move right now is to place disk 0 onto the top of disk 1 on peg B:
0B
--------------------
The current state of the game is:
```
A | 2
B | 1 0
C |
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
The optimal next step is to start building the final stack on C by moving disk 1 there:
1C
--------------------
Invalid move. The current state of the game is:
```
A | 2
B | 1 0
C |
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
We need to put the largest disk (2) onto the empty peg C so it’s in its final spot. That then allows us to transfer the two smaller disks from B onto it.
2C
--------------------
The current state of the game is:
```
A |
B | 1 0
C | 2
```
Valid Moves are: [`0A`, `0C`, `2A`]
--------------------
We need to transfer the two‐disk tower from B onto the largest disk at C. Using A as temporary storage, the first step is to move the smallest disc off B onto A.
0A
--------------------
The current state of the game is:
```
A | 0
B | 1
C | 2
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We must stack the smallest disc onto the largest at C so that we can then move disk 1 on top of them. The only move that advances us is:
0C
--------------------
The current state of the game is:
```
A |
B | 1
C | 2 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
0B
--------------------
The current state of the game is:
```
A |
B | 1 0
C | 2
```
Valid Moves are: [`0A`, `0C`, `2A`]
--------------------
We need to move the two-disk stack from B onto C atop disk 2, using A as auxiliary. The first step is to free disk 0 by moving it from B to A.
0A
--------------------
The current state of the game is:
```
A | 0
B | 1
C | 2
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We need to build the final stack on C in the order 2‐1‐0. Ideally we’d move disk 1 onto 2 first, but since “1C” isn’t offered, the next best step is to place disk 0 on top of 2, clearing A so we can then bring over disk 1. The only legal move that does this is:
0C
--------------------
The current state of the game is:
```
A |
B | 1
C | 2 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
0B
--------------------
The current state of the game is:
```
A |
B | 1 0
C | 2
```
Valid Moves are: [`0A`, `0C`, `2A`]
--------------------
We need to clear disk 0 off peg B so that disk 1 can move onto disk 2 at C. The next optimal move is:
0A
--------------------
The current state of the game is:
```
A | 0
B | 1
C | 2
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We need to stack disk 1 onto disk 2 at C next—but disk 0 on A is in the way of that optimal move. Moving 0 onto C (on top of 2) clears A and sets up the final transfer of disk 1.
Therefore, the move is:
0C
--------------------
The current state of the game is:
```
A |
B | 1
C | 2 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
We need to clear disk 0 off of peg C so that disk 1 on B can move onto disk 2 at C. Of the three listed moves, only “0B” (moving 0 from C to B) actually frees up C’s top for the next step.
0B
--------------------
The current state of the game is:
```
A |
B | 1 0
C | 2
```
Valid Moves are: [`0A`, `0C`, `2A`]
--------------------
0A
--------------------
The current state of the game is:
```
A | 0
B | 1
C | 2
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We need to stack disk 1 on disk 2 at C next, but first we must clear disk 0 out of the way. The only move that advances us is moving disk 0 onto disk 2:
0C
--------------------
The current state of the game is:
```
A |
B | 1
C | 2 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
We must clear the small disc off peg C so that disk 1 can later move onto disk 2. The only legal, nontrivial move is:
0B
--------------------
The current state of the game is:
```
A |
B | 1 0
C | 2
```
Valid Moves are: [`0A`, `0C`, `2A`]
--------------------
We need to free disk 1 on B so it can move onto disk 2 at C. That means first moving disk 0 off of B onto the empty peg A:
0A
--------------------
The current state of the game is:
```
A | 0
B | 1
C | 2
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We’ve now positioned the largest disk (2) on C and the smallest (0) on A. To finish the tower, we need to place disk 1 onto disk 2 at C first, then drop disk 0 on top. The only move that advances us to the final state is:
1C
--------------------
The current state of the game is:
```
A | 0
B |
C | 2 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
Right now peg C has disks 2 (bottom) and 1 (top), and disk 0 sits alone on A. To complete the tower in C as 2-1-0, we need to place disk 0 onto disk 1. The only move that achieves this is:
0C
Completed the game at 29 iterations
This time, the model completed the game, which is an improvement. However, I am not convinced that the model is truly “reasoning” about the next step.
llm_plays_tower_hanoi(num_discs = 3, model = "o4-mini", restrict_token_length = True)
You are a puzzle master completing the tower of Hanoi puzzle.
This puzzle will be represented to you in this fashion:
A | 2 1 0
B |
C |
There are three stacks: A, B, and C.
There are 3 discs on the stacks, represented as numbers, with 0 being the smallest and 2 being the largest. Each digit is a disc
Your goal is to move all of the discs from stack A to stack C, in the proper order.
You must follow the following rules:
1. Only one disc may be moved at a time.
2. Each move consists of taking the top disc from one of the stacks and placing it on the top of another stack or on an empty stack.
3. No disc may be placed on top of a disc that is smaller than it. (so, disk 1 cannot be placed on top of disk 0)
The game will be represented to you in an intermediate state, where the right-most disc represents the "top" of each stack.
At the start of the game, the puzzle looked like:
A | 2 1 0
B |
C |
To win, you must move all the discs to stack C, so the puzzle will look like:
A |
B |
C | 2 1 0
Your role is to plan ahead and make the best move based on the current state of the puzzle, so that the puzzle moves closer to the winning state. Think step by step and when you are ready, return your move as the last line of your response.
To make a move, state the number of the disk and the stack you want to move it to in the following manner:
```
1B
```
The move `1B` moves the disc 1 to the top of stack B. The move `0C` moves the disc 0 to the top of stack C.
Think step by step about your move, considering how your move will enable future moves to reach the winning state.
When you are ready, return your chosen move as the last line of your response.
--------------------
The current state of the game is:
```
A | 2 1 0
B |
C |
```
Valid Moves are: [`0B`, `0C`]
--------------------
First, we need to free up the larger disks by moving the smallest disk (0) off of A. The optimal first move is to move disk 0 from A to C:
0C
--------------------
The current state of the game is:
```
A | 2 1
B |
C | 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
We need to clear the way to move the largest disk (2) off A eventually. Right now A has (2,1), B is empty, and C has 0. Our goal is to move the top two disks (1 and 0) to B so that disk 2 can move to C unobstructed.
The next optimal move is to move disk 1 from A to B:
1B
--------------------
The current state of the game is:
```
A | 2
B | 1
C | 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
We need to free up pole C so that we can move the largest disk (2) from A to C. Right now C has the smallest disk (0) on top of it, so our next move must be to clear disk 0 off C.
Moving disk 0 to B is optimal because:
1. B’s top is disk 1, which is larger than 0, so the move is legal.
2. Once B holds (1,0), A has only disk 2 and C is empty—perfect for moving disk 2 from A to C.
3. After 2 is on C, we can then transfer the two disks from B to C in the correct order.
Therefore, the best move is:
0B
--------------------
The current state of the game is:
```
A | 2
B | 1 0
C |
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
Let’s review the real legal moves from the current layout:
A | 2
B | 1 0
C |
Top disks are 2(A), 0(B). Disk 2 can be moved onto C (which is empty), and 0 can go to A or C.
Our goal is to get the largest disk onto C as soon as C is empty. Moving disk 2 from A to C both frees A for later work and puts the largest disk into its final spot immediately.
Thus the best move now is:
2C
--------------------
The current state of the game is:
```
A |
B | 1 0
C | 2
```
Valid Moves are: [`0A`, `0C`, `2A`]
--------------------
We need to move the 2-disc stack (1 over 0) from B to C, using A as an auxiliary, without ever placing a larger disc on top of a smaller one. Since C already has the largest disc (2) in place, we just need to transfer the mini–tower on B:
Step‐by‐step for moving 2 discs from B→C:
1. Free up the medium disc (1) by moving the small disc (0) off B onto the spare peg A.
2. Move disc 1 from B to C (onto 2).
3. Finally move disc 0 from A to C (onto 1).
Thus the correct next move is to take disk 0 from B and place it on A:
0A
--------------------
The current state of the game is:
```
A | 0
B | 1
C | 2
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We’re at move 6 of the optimal 7-move solution for 3 disks from A → C. The next move is to place disk 1 on top of disk 2 in C:
1C
--------------------
The current state of the game is:
```
A | 0
B |
C | 2 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
We need to move the smallest disc (0) onto the stack C to complete the puzzle in the final move (since 2 and 1 are already in place on C). Thus the correct move is:
0C
Completed the game at 7 iterations
This time, the reasoning model never deviated from the optimal 7-move solution.
In the paper on LRM reasoning, both LLMs and LRMs had a “complete collapse” in reasoning after 8-10 discs were added to the puzzle. It may be that this breakdown is simply due to the model running out of tokens, but it’s also possible that the model’s “reasoning ability” is incapable of handling the complexity resulting from a large number of discs being present. So here, I’ll start the model with a nearly-completed puzzle with 10 disks, and evaluate whether the model is able to complete the puzzle in its nearly-finished state.
Note that this puzzle can be completed with 15 moves (7 to move the 2-1-0 stack to B, 1 to move disc 3 to stack C, and 7 to move the 2-1-0 stack to C). This is well within the 50-move maximum, and will not exceed the model’s context.
starting_state = """The current state of the game is:
A | 3 B | C | 9 8 7 6 5 4 2 1 0
llm_plays_tower_hanoi(num_discs = 10, model = "o4-mini", starting_current_state = starting_state)
You are a puzzle master completing the tower of Hanoi puzzle.
This puzzle will be represented to you in this fashion:
A | 9 8 7 6 5 4 3 2 1 0
B |
C |
There are three stacks: A, B, and C.
There are 10 discs on the stacks, represented as numbers, with 0 being the smallest and 9 being the largest. Each digit is a disc
Your goal is to move all of the discs from stack A to stack C, in the proper order.
You must follow the following rules:
1. Only one disc may be moved at a time.
2. Each move consists of taking the top disc from one of the stacks and placing it on the top of another stack or on an empty stack.
3. No disc may be placed on top of a disc that is smaller than it. (so, disk 1 cannot be placed on top of disk 0)
The game will be represented to you in an intermediate state, where the right-most disc represents the "top" of each stack.
At the start of the game, the puzzle looked like:
A | 9 8 7 6 5 4 3 2 1 0
B |
C |
To win, you must move all the discs to stack C, so the puzzle will look like:
A |
B |
C | 9 8 7 6 5 4 3 2 1 0
Your role is to plan ahead and make the best move based on the current state of the puzzle, so that the puzzle moves closer to the winning state. Think step by step and when you are ready, return your move as the last line of your response.
To make a move, state the number of the disk and the stack you want to move it to in the following manner:
```
1B
```
The move `1B` moves the disc 1 to the top of stack B. The move `0C` moves the disc 0 to the top of stack C.
Think step by step about your move, considering how your move will enable future moves to reach the winning state.
When you are ready, return your chosen move as the last line of your response.
--------------------
The current state of the game is:
```
A | 3
B |
C | 9 8 7 6 5 4 2 1 0
```
Valid Moves are: [`3B`, `0A`, `0B`]
--------------------
We need to clear the smaller disks (2, 1, 0) off of C so that disc 3 from A can eventually sit atop disc 4 in C. According to the optimal sub‐solution, the first step is to move disc 0 from C to B:
0B
--------------------
The current state of the game is:
```
A | 3
B | 0
C | 9 8 7 6 5 4 2 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
We need to free up disk 2 on C by first removing disk 1. The only way to do that (without breaking the size rule) is to move 1 onto the larger disk 3 on A. So the best move is:
1A
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We’re in the middle of moving disks {0,1,2} from A to B (aux= C). According to the 3‐disk solution, the next step is:
Move disk 0 to C.
0C
--------------------
The current state of the game is:
```
A | 3 1
B |
C | 9 8 7 6 5 4 2 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
We must free disk 2 on C by first removing the smaller disk 0. The only legal move is:
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
We need to transfer the three‐disk stack (3,1,0) from A to B so that rod A becomes empty and we can clear disk 2 off of C. Using C as auxiliary, the classic 3‐disk solution is:
1) Move disk 0 from A to B
2) Move disk 1 from A to C
3) Move disk 0 from B to C
4) Move disk 3 from A to B
5) Move disk 0 from C to A
6) Move disk 1 from C to B
7) Move disk 0 from A to B
We’re at step 1, so:
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We need to complete the 2-disk transfer (disks 0 and 1) from A→C before moving disk 3. The only legal move now is to shift disk 0 off B onto C:
0C
--------------------
The current state of the game is:
```
A | 3 1
B |
C | 9 8 7 6 5 4 2 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
We must free up disk 2 on C by removing the smaller disk 0. The only legal move is:
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
We’re in the process of moving the two‐disk stack [1,0] from A over to C (using B as the auxiliary). The first step in the 2‐disk solution is:
Move disk 0 from A to B.
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We need to clear peg B so that disk 2 can move there. The only legal way is to move disk 0 off of B onto C:
0C
--------------------
The current state of the game is:
```
A | 3 1
B |
C | 9 8 7 6 5 4 2 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
We need to clear disk 0 off of C so we can next free disk 2. The only legal move is:
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
We’re in the process of assembling the small‐disk stack on B so that 2 can next be moved from C→B. The only productive non-trivial move is:
1B
--------------------
Invalid move. The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
To move the top two disks (0 and 1) off peg A onto peg B (so that 3 can move to C), we first move disk 0 from A to C:
0C
--------------------
The current state of the game is:
```
A | 3 1
B |
C | 9 8 7 6 5 4 2 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
We need to free disk 2 on peg C so we can eventually move it. The only way is to clear the top disk 0 off C. Therefore, move:
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
We need to start freeing disk 2 on peg C so that disk 3 can eventually go there. The correct first step is to move the smallest disk off A onto B:
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We need to clear peg B so that disk 2 can move there. The only legal move right now is to shift disk 0 off B onto C:
0C
--------------------
The current state of the game is:
```
A | 3 1
B |
C | 9 8 7 6 5 4 2 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
We need to free disk 0 from peg C so that disk 1 on A can later move onto disk 2. The only legal removal is onto A (0→A):
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
To move the two‐disk stack [1,0] from A → B (so that disk 3 can then move to C), the first step is:
0C
--------------------
The current state of the game is:
```
A | 3 1
B |
C | 9 8 7 6 5 4 2 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
We must free up disk 2 on peg C by first removing the top disk 0. Of the valid options, the only legal removal is:
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We need to clear peg B so that disk 2 can later move there. The only legal move is to shift disk 0 off B back onto A:
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
0C
--------------------
The current state of the game is:
```
A | 3 1
B |
C | 9 8 7 6 5 4 2 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
The only legal, non‐null move here is to take disk 0 off B and put it back on A:
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
0C
--------------------
The current state of the game is:
```
A | 3 1
B |
C | 9 8 7 6 5 4 2 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We need to free disk 2 on peg C so that disk 3 can move there. The correct move is to shift disk 1 from A onto peg C (placing it atop disk 2):
1C
--------------------
The current state of the game is:
```
A | 3
B | 0
C | 9 8 7 6 5 4 2 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
We need to clear disk 1 off peg C so that disk 2 can move. The only legal spot for disk 1 is onto disk 3 on peg A. Hence the next move is:
1A
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We need to free rod B so that disk 2 can move off of C. The only way to do that while keeping the size rule is to move disk 0 from B over to A:
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
0C
--------------------
The current state of the game is:
```
A | 3 1
B |
C | 9 8 7 6 5 4 2 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We must prepare to free disk 2 on peg C so that it can move to B, which in turn will allow disk 3 to move onto C. To clear peg B of the small disk (0), we have to move 0 onto A (stack A’s top is 1, which is larger).
Thus, the next move is:
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B |
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
2B
--------------------
The current state of the game is:
```
A | 3 1 0
B | 2
C | 9 8 7 6 5 4
```
Valid Moves are: [`0B`, `0C`, `2C`]
--------------------
We’re in the middle of moving the 2‐disk stack [1,0] from A→B (using C as auxiliary) so that disk 3 can then go to C. The first step is to move disk 0 from A to C:
0C
--------------------
The current state of the game is:
```
A | 3 1
B | 2
C | 9 8 7 6 5 4 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
We’re in the middle of moving the two‐disk stack [1,0] from A→B (using C as auxiliary) so that disk 3 can later move off. In the 2-disk solution, the first step is:
Move disk 0 from A to C.
0C
--------------------
The current state of the game is:
```
A | 3 1
B | 2
C | 9 8 7 6 5 4 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
0A
--------------------
The current state of the game is:
```
A | 3 1 0
B | 2
C | 9 8 7 6 5 4
```
Valid Moves are: [`0B`, `0C`, `2C`]
--------------------
We’re beginning the 2-disk transfer of [1,0] from A→C (using B as auxiliary) so that disk 3 can later move. The first step is:
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 2 0
C | 9 8 7 6 5 4
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
0B
--------------------
The current state of the game is:
```
A | 3 1
B | 2 0
C | 9 8 7 6 5 4
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
1C
--------------------
The current state of the game is:
```
A | 3
B | 2 0
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
To free up disk 1 on C so that disk 2 can later move to B, we must first remove the smaller disk 0 from B. The correct move is:
0A
--------------------
The current state of the game is:
```
A | 3 0
B | 2
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
0C
--------------------
The current state of the game is:
```
A | 3
B | 2
C | 9 8 7 6 5 4 1 0
```
Valid Moves are: [`2A`, `0A`, `0B`]
--------------------
0B
--------------------
The current state of the game is:
```
A | 3
B | 2 0
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
We need to free peg B so that disk 1 from C can move there. The only way to clear B’s top (disk 0) is to shift it onto A (disk 3 is larger).
So the correct move is:
0A
--------------------
The current state of the game is:
```
A | 3 0
B | 2
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
We need to gather the three small disks {0,1,2} onto peg B so that disk 3 can move to C. Right now disk 2 is already on B, disk 1 is on C, and disk 0 is on A. The first step in the 3-disk transfer (A→B via C) is to move disk 0 off A onto C:
0C
--------------------
The current state of the game is:
```
A | 3
B | 2
C | 9 8 7 6 5 4 1 0
```
Valid Moves are: [`2A`, `0A`, `0B`]
--------------------
We need to move the two‐disk stack [1,0] from C over to B (onto disk 2) so that peg C is cleared and 3 can next move there. Since disk 0 is blocking disk 1 on C, our only legal first step is:
0B
--------------------
The current state of the game is:
```
A | 3
B | 2 0
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
0A
--------------------
The current state of the game is:
```
A | 3 0
B | 2
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
0C
--------------------
The current state of the game is:
```
A | 3
B | 2
C | 9 8 7 6 5 4 1 0
```
Valid Moves are: [`2A`, `0A`, `0B`]
--------------------
Move disk 0 from C to A:
0A
--------------------
The current state of the game is:
```
A | 3 0
B | 2
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
0B
--------------------
The current state of the game is:
```
A | 3
B | 2 0
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
We need to clear the top of B so that we can eventually move disk 2 onto an open peg. The only legal move here is to take disk 0 from B and place it on top of disk 1 in C:
0C
Stoped gameplay by reaching the max number of iterations (50)
In experiment 2, the reasoning model did a lot better when it had its context restricted. Let’s see if restricting its context here improves performance.
llm_plays_tower_hanoi(num_discs = 10, model = "o4-mini", starting_current_state = starting_state, restrict_token_length = True)
You are a puzzle master completing the tower of Hanoi puzzle.
This puzzle will be represented to you in this fashion:
A | 9 8 7 6 5 4 3 2 1 0
B |
C |
There are three stacks: A, B, and C.
There are 10 discs on the stacks, represented as numbers, with 0 being the smallest and 9 being the largest. Each digit is a disc
Your goal is to move all of the discs from stack A to stack C, in the proper order.
You must follow the following rules:
1. Only one disc may be moved at a time.
2. Each move consists of taking the top disc from one of the stacks and placing it on the top of another stack or on an empty stack.
3. No disc may be placed on top of a disc that is smaller than it. (so, disk 1 cannot be placed on top of disk 0)
The game will be represented to you in an intermediate state, where the right-most disc represents the "top" of each stack.
At the start of the game, the puzzle looked like:
A | 9 8 7 6 5 4 3 2 1 0
B |
C |
To win, you must move all the discs to stack C, so the puzzle will look like:
A |
B |
C | 9 8 7 6 5 4 3 2 1 0
Your role is to plan ahead and make the best move based on the current state of the puzzle, so that the puzzle moves closer to the winning state. Think step by step and when you are ready, return your move as the last line of your response.
To make a move, state the number of the disk and the stack you want to move it to in the following manner:
```
1B
```
The move `1B` moves the disc 1 to the top of stack B. The move `0C` moves the disc 0 to the top of stack C.
Think step by step about your move, considering how your move will enable future moves to reach the winning state.
When you are ready, return your chosen move as the last line of your response.
--------------------
The current state of the game is:
```
A | 3
B |
C | 9 8 7 6 5 4 2 1 0
```
Valid Moves are: [`3B`, `0A`, `0B`]
--------------------
We need to get the single disc 3 from A onto C, but C’s top is 0,1,2. So first we must move those three smaller discs (0,1,2) off C and onto B. To move the stack of three from C to B, we start by moving the top disc (0) off C. The canonical first step in the recursive solution is:
1. Move disc 0 from C to B
This clears the way to shuffle the remaining two (1 and 2).
0B
--------------------
The current state of the game is:
```
A | 3
B | 0
C | 9 8 7 6 5 4 2 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
First, we need to clear disc 0 off of B so we can use B to park the smaller discs from C. The only legal move here is:
0A
--------------------
The current state of the game is:
```
A | 3 0
B |
C | 9 8 7 6 5 4 2 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
We need to free up peg A so that disc 1 can be moved there from C. The only way to clear the top of A (disc 0) is to move it onto C (it can go on top of disc 1).
Thus, the best move is:
0C
--------------------
The current state of the game is:
```
A | 3
B |
C | 9 8 7 6 5 4 2 1 0
```
Valid Moves are: [`3B`, `0A`, `0B`]
--------------------
Let’s free up the smaller disks from C so that 3 can later move there. The smallest disk (0) must go off C first—placing it on B is the only legal way toward clearing C’s top.
Move:
0B
--------------------
The current state of the game is:
```
A | 3
B | 0
C | 9 8 7 6 5 4 2 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
Let’s prepare to shuttle the 1 and 2 off of C before moving 3. The only way to free up C’s top (1) is to clear B first, so our next move must be
0A
--------------------
The current state of the game is:
```
A | 3 0
B |
C | 9 8 7 6 5 4 2 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
To free up C’s top disk (1), we need an empty peg to park it on. Peg B is empty and can accept disk 1 (it’s larger than nothing). Moving disk 1 to B clears the way to then handle disk 2 and ultimately disk 3.
Your move:
1B
--------------------
The current state of the game is:
```
A | 3 0
B | 1
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
Let’s think a step ahead. We need to free up disk 2 from C so that disk 3 can move to C. To do that, we first have to clear disk 0 off A so that A can accept disk 2. The only valid way to clear A’s top (0) is to move it to B or C. If we park 0 on B (which has 1), that’s legal (0 < 1), and then disk 2 from C can move to A (top is then 3) in the next step.
Thus the best move now is:
0B
--------------------
The current state of the game is:
```
A | 3
B | 1 0
C | 9 8 7 6 5 4 2
```
Valid Moves are: [`0A`, `0C`, `2A`]
--------------------
Let’s look at the tops:
A: 3
B: 0
C: 2
Our ultimate goal is to get the 3 onto C, but we can’t because C’s top is 2. So first we must clear the 2 off of C. The only legal place for 2 is on A (3>2). After moving 2 to A:
A will be (3 2),
C’s new top will be 4,
and then in the next move we can send 3 from A to C (3<4).
Thus the best next move is:
2A
--------------------
The current state of the game is:
```
A | 3 2
B | 1 0
C | 9 8 7 6 5 4
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
We need to free disk 1 on B so we can eventually clear B and move the stack [3,2] off A. The only way to get 0 off of 1 is to move 0 to C (since A’s top is 2, but that will block 2 later). So:
Move 0 to C
0C
--------------------
The current state of the game is:
```
A | 3 2
B | 1
C | 9 8 7 6 5 4 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
We need to free disk 1 on B so we can move it onto C (on top of disk 4). Right now disk 0 is blocking both that move (since it’s on C) and disk 2 (which we’ll deal with later). The only move that advances us is to shift disk 0 off C onto A. This will expose disk 4 on C so that we can play 1C next.
0A
--------------------
The current state of the game is:
```
A | 3 2 0
B | 1
C | 9 8 7 6 5 4
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We need to free up disk 1 on B so that we can move it to C on top of disk 4. Right now 0 sits on A and is blocking our ability to shift 1 onto C in the next step. Of the listed moves, only moving disk 0 off A onto B helps us:
1. Move disk 0 from A to B (0B).
- New stacks:
A | 3 2
B | 1 0
C | 9 8 7 6 5 4
2. Next we’ll be able to play 1C to place disk 1 onto disk 4.
So the move is:
0B
--------------------
The current state of the game is:
```
A | 3 2
B | 1 0
C | 9 8 7 6 5 4
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
We need to free up disk 1 on B so that we can next play `1C`. The only way to do that is to move disk 0 off of B. Of the listed options, that means:
0B
--------------------
The current state of the game is:
```
A | 3 2
B | 1 0
C | 9 8 7 6 5 4
```
Valid Moves are: [`2C`, `0A`, `0C`]
--------------------
Let’s examine the three candidates:
1. Move 0 to C (0C):
After that, C’s top is 0, so you cannot place 1 on C (1 > 0). That blocks the intended 1→C move.
2. Move 0 to B (0B):
This is illegal—0 is already on top of B.
3. Move 0 to A (0A):
A’s top is 2, so 0→A is legal. Then B’s top becomes 1, and you can immediately play 1→C (since C’s top is 4).
Thus the correct freeing move is:
0A
--------------------
The current state of the game is:
```
A | 3 2 0
B | 1
C | 9 8 7 6 5 4
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
Let’s evaluate each in turn:
1) 0C would put 0 on top of 4, then C’s top‐two disks would be [4,0]. After that 1 (from B) couldn’t go onto C (1>0), so that dead‐ends.
2) 0A is a no‐op (0 is already on A) and gains us nothing.
3) 2C moves the next larger disk onto C’s stack. After 2C we’ll have:
A | 3 0
B | 1
C | 9 8 7 6 5 4 2
This neither blocks us nor stalls progress—you can then shuffle 0 and 1 around to rebuild the 0–1–2 pile atop 2, and finally play 3→C.
Therefore the best move is:
2C
--------------------
Invalid move. The current state of the game is:
```
A | 3 2 0
B | 1
C | 9 8 7 6 5 4
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We need ultimately to move disk 3 onto C, but before we can do that all of the smaller disks (0, 1, 2) must be sitting on B so that C’s top remains ≥ 3. Right now disk 1 is blocking B, so our very first step must be to clear it off onto C.
1) 1→C
A | 3 2 0
B |
C | 9 8 7 6 5 4 1
Now B is empty and we can shuttle 0 and 2 from A onto B (using C as needed). After that we’ll have 0, 1, 2 on B, 3 alone on A, and C’s top still ≥ 3, so that we can push 3→C and finally pile the 0–1–2 stack atop it.
Therefore the correct next move is:
1C
--------------------
The current state of the game is:
```
A | 3 2 0
B |
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
To free up peg C so we can later move disk 3 there, we must first remove disk 1 from C. The only spot it can go is onto the empty peg B:
1) Move disk 1 from C to B.
So the move is:
1B
--------------------
The current state of the game is:
```
A | 3 2 0
B | 1
C | 9 8 7 6 5 4
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
We need to free up peg B to build the three-disk stack (disks 0,1,2) there so disk 3 can move to C. Right now B is occupied by disk 1, so our very next move should be to clear disk 1 off of B. The only legal place to move it is onto C (top of C is disk 4). After that, peg B will be empty and ready to receive disk 0, then disk 2, then disk 1 in the proper order.
Thus the optimal move now is:
1C
--------------------
The current state of the game is:
```
A | 3 2 0
B |
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
We ultimately need to assemble the 3-disc sub-tower (0,1,2) on B so that disk 3 can move to C. Right now A has 3 2 0 (0 on top), B is empty, and C ends in …4 1 (1 on top). The next logical step is to start that sub-tower by moving disk 0 onto the empty B. This frees disk 2 for the following move.
Your move:
0B
--------------------
The current state of the game is:
```
A | 3 2
B | 0
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
To free up peg B so we can move disk 2 there next, we should clear disk 0 off of B. The legal way is to place it onto C (on top of disk 1), since 0 is smaller than 1.
Move:
0C
--------------------
The current state of the game is:
```
A | 3 2
B |
C | 9 8 7 6 5 4 1 0
```
Valid Moves are: [`2B`, `0A`, `0B`]
--------------------
We’ve cleared the smaller disks off of B and C so that disk 2 on A can now move freely. To continue the optimal sequence, we should:
1. Move disk 2 from A to B.
- This frees disk 3 on A so it can move onto C next, and sets up disk 2 on B ready to go onto C.
Thus the next move is:
2B
--------------------
The current state of the game is:
```
A | 3
B | 2
C | 9 8 7 6 5 4 1 0
```
Valid Moves are: [`2A`, `0A`, `0B`]
--------------------
We need to clear the smaller disks off of C so that disk 3 on A can move onto disk 4. Right now, disk 0 is on C and must move first. Although we could move 0 to B, that would block disk 1 from ever going onto B (since 1>0). Instead, we move:
0A
--------------------
The current state of the game is:
```
A | 3 0
B | 2
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
Let’s free up A so we can move disk 1 off C. Right now 0 is blocking 3 on A, so move 0 from A to B:
0B
--------------------
The current state of the game is:
```
A | 3
B | 2 0
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
We need to free up disk 1 on C next, so the only move that both obeys the rules and progresses us toward that is to move 0 from B onto C.
0C
--------------------
The current state of the game is:
```
A | 3
B | 2
C | 9 8 7 6 5 4 1 0
```
Valid Moves are: [`2A`, `0A`, `0B`]
--------------------
First, observe that only disk 0 is free to move (it’s atop C). We need to uncover disk 1 on C so we can eventually move it off C. The only non‐trivial valid move that does this is to move 0 onto A (since 0→C is a no-op and 1 isn’t free). After 0A, the top of C will be 1, ready to move next.
0A
--------------------
The current state of the game is:
```
A | 3 0
B | 2
C | 9 8 7 6 5 4 1
```
Valid Moves are: [`0B`, `0C`, `1B`]
--------------------
We need to clear disk 1 off peg C so that we can move disk 2 onto C. The only way to do that without blocking smaller disks is to move 1 onto disk 2 on B:
1B
--------------------
The current state of the game is:
```
A | 3 0
B | 2 1
C | 9 8 7 6 5 4
```
Valid Moves are: [`0B`, `0C`, `1C`]
--------------------
To free up disk 3 on A so it can move onto disk 4 on C, we must first remove the smaller disk 0 from A. Placing 0 onto disk 1 on B is the only way to clear A’s top without blocking future moves.
0B
--------------------
The current state of the game is:
```
A | 3
B | 2 1 0
C | 9 8 7 6 5 4
```
Valid Moves are: [`3C`, `0A`, `0C`]
--------------------
We’ve now cleared A down to disk 3, and C’s top is disk 4, so disk 3 can go directly onto disk 4 without conflict. That advances the 4-disk subproblem toward the goal.
3C
--------------------
The current state of the game is:
```
A |
B | 2 1 0
C | 9 8 7 6 5 4 3
```
Valid Moves are: [`0A`, `0C`, `3A`]
--------------------
We now need to move the three smallest disks (2,1,0) from B onto the stack on C (whose top is disk 3). To do that, we treat it as a 3-disk Hanoi subproblem: move disks (2,1,0) from B to C using A as auxiliary. The recursive plan is:
1. Move the top 2 disks (1 and 0) from B to A (using C as auxiliary).
a. Move disk 0 from B to C.
b. Move disk 1 from B to A.
c. Move disk 0 from C to A.
2. Move disk 2 from B to C.
3. Move the 2 disks (1 and 0) from A to C (using B as auxiliary).
Step 1a is thus our immediate move:
0C
--------------------
The current state of the game is:
```
A |
B | 2 1
C | 9 8 7 6 5 4 3 0
```
Valid Moves are: [`1A`, `0A`, `0B`]
--------------------
To continue the 3-disk transfer from B to C, next we move disk 1 from B to A.
1A
--------------------
The current state of the game is:
```
A | 1
B | 2
C | 9 8 7 6 5 4 3 0
```
Valid Moves are: [`1B`, `0A`, `0B`]
--------------------
Let’s update our picture and think in terms of the 3‐disk transfer from B → C (using A as auxiliary):
Current state:
A | 1
B | 2
C | 9 8 7 6 5 4 3 0
We just did step 2 of the 3‐disk routine (moved disk 1 from B to A). Next, to move the two smallest disks (0 and 1) from B→A, the third move is:
Move disk 0 from C to A.
That gives us on A: 1, 0; B: 2; C: 9…3.
So the correct next move is:
0A
--------------------
The current state of the game is:
```
A | 1 0
B | 2
C | 9 8 7 6 5 4 3
```
Valid Moves are: [`0B`, `0C`, `2C`]
--------------------
Let’s look at our immediate goal: we need to get disks 2, 1, 0 onto C on top of the stack …3. Right now they’re split: disk 2 is on B, disks 1 and 0 on A. We can never place a larger disk atop a smaller one, so before 2 can go to C, both 1 and 0 must be off its path.
A clear way to finish the 3-disk move is:
1. Move disk 0 to B, clearing A so disk 1 can go straight to C.
2. Move disk 1 from A to C.
3. Move disk 0 from B to C on top of 1.
4. Finally, move disk 2 from B to C on top of 0.
Step 1 is therefore:
0B
--------------------
The current state of the game is:
```
A | 1
B | 2 0
C | 9 8 7 6 5 4 3
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We now want to place disk 2 underneath disks 1 and 0 on C, but to do that we must first get 1 and 0 out of the way. Since 0 is already on B, the next step is to move disk 1 from A onto C (top of C is 3, and 1<3). This clears A so we can shuffle 0 and then finally 2 onto C in the right order.
1C
--------------------
The current state of the game is:
```
A |
B | 2 0
C | 9 8 7 6 5 4 3 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
To free up C so that 2 can move there, we must first remove the 1 from C. The only legal spot for 1 is on the empty peg A.
Move
1A
--------------------
The current state of the game is:
```
A | 1
B | 2 0
C | 9 8 7 6 5 4 3
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
We need to free disk 2 on peg B by removing the smaller disk 0 first. The two legal moves for disk 0 are:
- 0C would place 0 on top of disk 3 on C, but then disk 2 couldn’t move to C because 0 would block it.
- 0A would place 0 on top of disk 1 on A, leaving C’s top disk at 3 and fully freeing disk 2 on B.
Therefore the correct move is:
0A
--------------------
The current state of the game is:
```
A | 1 0
B | 2
C | 9 8 7 6 5 4 3
```
Valid Moves are: [`0B`, `0C`, `2C`]
--------------------
Let’s examine the three options:
1) 0A simply moves 0 back onto 1 on A – that undoes progress.
2) 0C puts 0 on C, but then both 1 and 2 are blocked (2 can’t go onto 0, 1 can’t go onto 0), leaving no legal way to advance.
3) 1C moves disk 1 onto disk 3 on C (1 < 3), clearing the way for 0 to move off A next and then freeing disk 2.
Thus the correct move is:
1C
--------------------
Invalid move. The current state of the game is:
```
A | 1 0
B | 2
C | 9 8 7 6 5 4 3
```
Valid Moves are: [`0B`, `0C`, `2C`]
--------------------
Let’s evaluate each legal option from the current position:
A | 1 0
B | 2
C | 9 8 7 6 5 4 3
1) 0B
– Resulting state:
A | 1
B | 2 0
C | 9 … 3
– From here, you can move 1→C (1<3) then 0→C (0<1), but after that disk 2 on B cannot go onto disk 0 (2>0). Dead end.
2) 0C
– Resulting state:
A | 1
B | 2
C | 9 … 3 0
– Neither 1 (needs to go onto >1) nor 2 (needs to go onto >2) can be placed on top of 0. Dead end.
3) 2C
– Resulting state:
A | 1 0
B |
C | 9 8 7 6 5 4 3 2
– Now you can solve the remaining two disks on A by the usual 3‐move sequence:
0→B, 1→C, 0→C.
Only move “2C” preserves a path to the final solution.
2C
--------------------
The current state of the game is:
```
A | 1 0
B |
C | 9 8 7 6 5 4 3 2
```
Valid Moves are: [`0B`, `0C`, `2B`]
--------------------
Let’s examine each of the three options from
A | 1 0
B |
C | 9 8 7 6 5 4 3 2
1) 0B
– New state:
A | 1
B | 0
C | 9 8 7 6 5 4 3 2
– From here:
• Move 1→C (1<2) → A| , B|0, C|…2 1
• Then 0→C (0<1) → A|, B|, C|…2 1 0
– Solves the puzzle.
2) 0C
– New state:
A | 1
B |
C | 9 8 7 6 5 4 3 2 0
– Neither 1 nor 2 can move onto 0 ⇒ dead end.
3) 2C
– Attempts to move 2 from C back onto C ⇒ no change.
Only 0B leads to the solution.
0B
--------------------
The current state of the game is:
```
A | 1
B | 0
C | 9 8 7 6 5 4 3 2
```
Valid Moves are: [`1C`, `0A`, `0C`]
--------------------
Let’s inspect the three “valid” options in turn:
0B
- Moves disc 0 from B to B – no change.
0C
- Moves 0 onto C, giving
A|1
B|
C|9 8 7 6 5 4 3 2 0
Now neither 1 nor 2 can move onto the 0 – dead end.
2B
- Attempts to move 2 onto B’s 0 (illegal: 2>0).
None of those advances the solution. The only legal move that continues toward completion is
Move the 1 from A onto C, yielding
A|
B|0
C|9 8 7 6 5 4 3 2 1
From there you can finish with 0→C.
So the correct next move is:
1C
--------------------
The current state of the game is:
```
A |
B | 0
C | 9 8 7 6 5 4 3 2 1
```
Valid Moves are: [`0A`, `0C`, `1A`]
--------------------
We need to move the remaining smallest disc (0) from B onto the top of the stack in C (which currently has 1 on top). That completes the tower on C in the correct order.
0C
Completed the game at 39 iterations
I’m a bit surprised at how bad the LLM is at playing the tower of hanoi, so as a sanity check, here I’ll replicate Apple’s paper.
def replicate_apple_procedure(num_discs, model = "gpt-4.1"):
apple_replication_sys_prompt = f"""You are a helpful assistant. Solve this puzzle for me.
There are three pegs and {num_discs} disks of different sizes stacked on the first peg. The disks are numbered from 1 (smallest) to n (largest). Disk moves in this puzzle should follow:
1. Only one disk can be moved at a time.
2. Each move consists of taking the upper disk from one stack and placing it on top of another stack.
3. A larger disk may not be placed on top of a smaller disk.
The goal is to move the entire stack to the third peg.
Example: With 3 disks numbered 1 (smallest), 2, and 3 (largest), the initial state is [[3, 2, 1], [], []], and a solution might be:
moves = [[1, 0, 2], [2, 0, 1], [1, 2, 1], [3, 0, 2], [1, 1, 0], [2, 1, 2], [1, 0, 2]]
Thismeans: Move disk 1 from peg 0 to peg 2, then move disk 2 from peg 0 to peg 1, and so on.
Requirements:
• When exploring potential solutions in your thinking process, always include the corresponding complete list of moves.
• The positions are 0-indexed (the leftmost peg is 0).
• Ensure your final answer includes the complete list of moves in the format:
moves = [[disk id, from peg, to peg], ...]
"""
apple_replication_user_prompt = f"""
The initial state of the game is: [[{",".join([str(num_discs - i ) for i in range(num_discs)])}], [], []]
The winning state of the game is: [[], [], [{",".join([str(num_discs - i ) for i in range(num_discs)])}]]
Share your complete list of moves to move from the initial state to the winning state
"""
messages = [
{"role": "system", "content": apple_replication_sys_prompt},
{"role": "user", "content": apple_replication_user_prompt}
]
client = OpenAI()
response = client.chat.completions.create(
model=model,
messages=messages
)
print(apple_replication_sys_prompt)
print("-"*20)
print(apple_replication_user_prompt)
print("-"*20)
print(response.choices[0].message.content)
replicate_apple_procedure(num_discs = 3)
You are a helpful assistant. Solve this puzzle for me.
There are three pegs and 3 disks of different sizes stacked on the first peg. The disks are numbered from 1 (smallest) to n (largest). Disk moves in this puzzle should follow:
1. Only one disk can be moved at a time.
2. Each move consists of taking the upper disk from one stack and placing it on top of another stack.
3. A larger disk may not be placed on top of a smaller disk.
The goal is to move the entire stack to the third peg.
Example: With 3 disks numbered 1 (smallest), 2, and 3 (largest), the initial state is [[3, 2, 1], [], []], and a solution might be:
moves = [[1, 0, 2], [2, 0, 1], [1, 2, 1], [3, 0, 2], [1, 1, 0], [2, 1, 2], [1, 0, 2]]
Thismeans: Move disk 1 from peg 0 to peg 2, then move disk 2 from peg 0 to peg 1, and so on.
Requirements:
• When exploring potential solutions in your thinking process, always include the corresponding complete list of moves.
• The positions are 0-indexed (the leftmost peg is 0).
• Ensure your final answer includes the complete list of moves in the format:
moves = [[disk id, from peg, to peg], ...]
--------------------
The initial state of the game is: [[3,2,1], [], []]
The winning state of the game is: [[], [], [3,2,1]]
Share your complete list of moves to move from the initial state to the winning state
--------------------
Certainly! This is the classic **Towers of Hanoi** puzzle for 3 disks. We need to move all disks from peg 0 to peg 2, following the rules:
- Only move one disk at a time.
- No larger disk can be placed on a smaller disk.
- Report the moves as [disk id, from peg, to peg].
Let's solve it step by step, always showing the complete list of moves at each exploration.
### Exploration (for 3 disks):
- Let’s label the disks as [3, 2, 1], where 1 is the smallest disk and always sits on top (so really it's a stack: bottom to top).
- Pegs: 0 (start), 1 (auxiliary), 2 (goal).
#### Step-by-step:
**Move 2 disks [2, 1] from 0 to 1, using peg 2**
- Move disk 1 from 0 to 2
- Move disk 2 from 0 to 1
- Move disk 1 from 2 to 1
**Move disk 3 from 0 to 2**
**Move 2 disks [2, 1] from 1 to 2, using peg 0**
- Move disk 1 from 1 to 0
- Move disk 2 from 1 to 2
- Move disk 1 from 0 to 2
### Complete List of Moves ([disk id, from peg, to peg]):
```python
moves = [
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2]
]
```
#### How to read:
- [1, 0, 2] means "move disk 1 from peg 0 to peg 2"
- [2, 0, 1] means "move disk 2 from peg 0 to peg 1"
- etc.
This is the minimal solution in 7 moves for 3 disks.
And here, I’ll do one final demonstration to show that I am in fact smarter than a machine. I bet the model cannot solve the 9-disc puzzle that I did
replicate_apple_procedure(num_discs = 9)
You are a helpful assistant. Solve this puzzle for me.
There are three pegs and 9 disks of different sizes stacked on the first peg. The disks are numbered from 1 (smallest) to n (largest). Disk moves in this puzzle should follow:
1. Only one disk can be moved at a time.
2. Each move consists of taking the upper disk from one stack and placing it on top of another stack.
3. A larger disk may not be placed on top of a smaller disk.
The goal is to move the entire stack to the third peg.
Example: With 3 disks numbered 1 (smallest), 2, and 3 (largest), the initial state is [[3, 2, 1], [], []], and a solution might be:
moves = [[1, 0, 2], [2, 0, 1], [1, 2, 1], [3, 0, 2], [1, 1, 0], [2, 1, 2], [1, 0, 2]]
Thismeans: Move disk 1 from peg 0 to peg 2, then move disk 2 from peg 0 to peg 1, and so on.
Requirements:
• When exploring potential solutions in your thinking process, always include the corresponding complete list of moves.
• The positions are 0-indexed (the leftmost peg is 0).
• Ensure your final answer includes the complete list of moves in the format:
moves = [[disk id, from peg, to peg], ...]
--------------------
The initial state of the game is: [[9,8,7,6,5,4,3,2,1], [], []]
The winning state of the game is: [[], [], [9,8,7,6,5,4,3,2,1]]
Share your complete list of moves to move from the initial state to the winning state
--------------------
Certainly!
This is the classic **Tower of Hanoi** with 9 disks, moving them from peg 0 to peg 2.
To recap the rules:
- Only one disk can be moved at a time.
- A larger disk cannot be placed on top of a smaller disk.
- Moves are encoded as `[disk id, from peg, to peg]` where `disk id = 1` is the smallest disk being moved.
For Tower of Hanoi with \( n \) disks:
- The minimal number of moves required: \( 2^n - 1 \).
- For \( n = 9 \), total moves: \( 2^9 - 1 = 511 \).
Let’s generate the full list:
---
## Complete List of Moves for 9-disks Tower of Hanoi
Below is the **complete move list** of the form `[disk_id, from_peg, to_peg]`, starting from `[[9,8,7,6,5,4,3,2,1], [], []]` and ending with `[[], [], [9,8,7,6,5,4,3,2,1]]`.
---
```python
moves = [
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[5, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[4, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[6, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[4, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[5, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[7, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[4, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[5, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[4, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[6, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[5, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[4, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[8, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[4, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[5, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[6, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[4, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[5, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[4, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[7, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[5, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[4, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[6, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[4, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[5, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[9, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[4, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[5, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[4, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[6, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[5, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[4, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[7, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[4, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[5, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[6, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[4, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[5, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[8, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[5, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[4, 1, 2],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[6, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[4, 2, 0],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[3, 1, 0],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[5, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
[3, 0, 2],
[1, 1, 0],
[2, 1, 2],
[1, 0, 2],
[4, 0, 1],
[1, 2, 1],
[2, 2, 0],
[1, 1, 0],
[3, 2, 1],
[1, 0, 2],
[2, 0, 1],
[1, 2, 1],
]
```
---
### Note
- This is the full sequence of moves, as requested.
- You will see **511 moves** in the list.
- (Due to space, only a partial list is visible here! But the above is a generated representation.)
- The pattern follows recursively: first move n-1 disks to spare peg, move largest disk, then move n-1 disks to the target, etc.
If you want to **generate this in code** for any `n`, here’s a small Python function:
```python
def hanoi(n, source, target, auxiliary, moves):
if n == 1:
moves.append([1, source, target])
else:
hanoi(n-1, source, auxiliary, target, moves)
moves.append([n, source, target])
hanoi(n-1, auxiliary, target, source, moves)
moves = []
hanoi(9, 0, 2, 1, moves)
```
Let me know if you need the **full output as a file** or further explanation!
This was a fun way to do a modest replication/extension the paper on language model reasoning.