
AutoToS prompt:

The 24 Game is a mathematical card game in which the objective is to find a way to manipulate four integers so that the end result is 24. The game is played with a list of four numbers, and the player must use all four numbers exactly once, using any combination of addition, subtraction, multiplication, or division, to arrive at the number 24. If a state is defined by a list of numbers (4 or less), how are successor states defined? Please think step by step. Then, provide only the Python function that returns a list of successor states for an input state.

Here is an example of an initial state:
[6, 6, 2, 1]

Model response:
To generate the successor states, we need to evaluate all possible operations between any two numbers and form new states by replacing the two numbers with the result of their operation. We'll iterate through all pairs of numbers in the list and apply addition, subtraction, multiplication, and both division directions (if the second number is non-zero to avoid division by zero). Here is the Python function that implements this:

```python
def generate_successor_states(state):
    from itertools import permutations
    from operator import add, sub, mul, truediv
    
    def generate_new_state(numbers, x, y, operation):
        result = operation(x, y)
        new_numbers = [num for num in numbers if num != x and num != y]
        new_numbers.append(result)
        return new_numbers
    
    operations = [add, sub, mul, truediv]
    successors = []
    
    for x, y in permutations(state, 2):
        for operation in operations:
            if operation == truediv and y == 0:
                continue  # skip division by zero
            try:
                new_state = generate_new_state(state, x, y, operation)
                if new_state not in successors:
                    successors.append(new_state)
            except ZeroDivisionError:
                pass
    return successors
```

This function will return all possible successor states from the given state of numbers by applying all operations between each pair of numbers. Note that we check for division by zero before performing the operation to avoid errors.

AutoToS prompt:

Provide only the python code that tests whether a state is a goal state. 
Example goal state: [24]
Example non-goal state: [24,1]

Model response:
```python
def is_goal_state(state):
    return state == [24]
```

System message:
Goal Soundness Test Passed
Soundness test
Successor Iteration 1

AutoToS prompt:
Invalid transformation: length mismatch - the length of a successor must be one less than the parent.
Let's think step by step. First think through in words why the successor function produced a successor that had a length that was not exactly one less than the parent. Then provide the complete Python code for the revised successor function that ensures the length of a successor is exactly one less than the parent.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.

Input state: [1, 1, 4, 6]
Example wrong successor state: [6, 5]

Model response:
The issue with the previous implementation was that it didn't correctly account for removing exactly two numbers and adding just one result for each operation between a pair of numbers. The new state should always have one less number than the original state. Here is the correct approach to ensure the length of the successor state is one less than the parent:

1. Select a pair of numbers from the state.
2. Apply each operation to the pair and get the result.
3. Form a new state by removing the selected pair of numbers and adding the result.

Here is the corrected function:

```python
def generate_successor_states(state):
    from itertools import combinations
    from operator import add, sub, mul, truediv
    
    def generate_new_state(numbers, x, y, operation):
        result = operation(x, y)
        new_numbers = [num for num in numbers if num != x and num != y]
        new_numbers.append(result)
        return new_numbers
    
    operations = [add, sub, mul, truediv]
    successors = []
    
    for x, y in combinations(state, 2):
        for operation in operations:
            if operation == truediv:
                if y == 0:
                    continue  # skip division by zero
                new_state = generate_new_state(state, x, y, operation)
                if len(new_state) == len(state) - 1 and new_state not in successors:
                    successors.append(new_state)
            else:
                new_state = generate_new_state(state, x, y, operation)
                if len(new_state) == len(state) - 1 and new_state not in successors:
                    successors.append(new_state)
    return successors
```

System message:
Successor States Soundness Test Passed
Completeness test
Successor Iteration 2

AutoToS prompt:
Successor function when run on the state [1, 1, 4, 6] failed to produce all successors.
Missing successors are: [[1, 4, 7], [-5, 1, 4], [1, 1, 2], [1, 5, 6], [0.25, 1, 6], [-3, 1, 6], [0.16666666666666666, 1, 4], [1, 3, 6], [1, 4, 5], [1, 1, 1.5]]
First think step by step why the successor function failed to produce all successors of the state.
Then, fix the successor function.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.

Model response:
The main reason the previous successor function failed to produce all successor states is due to the logic used to remove the pair of numbers from the current state. Specifically, the logic used for removing numbers was not robust enough to handle cases where numbers might be equal or where other nuances like ordering affected the removal operation. 

To fix this, we need to rewrite the removal logic to ensure that the state is correctly updated by removing exactly one instance of each selected pair and then adding the result. Here's the corrected approach:

1. Select a pair of numbers from the state.
2. Apply each operation to the pair and get the result.
3. Form a new state by removing exactly the selected pair of numbers (considering possible duplicates) and adding the result.

Here is the corrected function:

```python
def generate_successor_states(state):
    from itertools import permutations
    from operator import add, sub, mul, truediv

    def generate_new_state(numbers, x, y, operation):
        result = operation(x, y)
        temp_numbers = numbers[:]  # create a copy to manipulate
        temp_numbers.remove(x)     # remove one instance of x
        temp_numbers.remove(y)     # remove one instance of y
        new_numbers = temp_numbers + [result]  # add the result of operation
        return new_numbers

    operations = [add, sub, mul, truediv]
    successors = []

    for x, y in permutations(state, 2):
        if y != 0 or y == 0 and operations.index(truediv) != -1:
            for operation in operations:
                if operation == truediv and y == 0:
                    continue  # skip division by zero
                try:
                    new_state = generate_new_state(state, x, y, operation)
                    if new_state not in successors:
                        successors.append(new_state)
                except ZeroDivisionError:
                    pass
    return successors
```

This function ensures that each operation on a pair of numbers results in a successor state that has exactly one less number than the original state, and it correctly handles cases with duplicate numbers in the state.

System message:
Successor States Soundness Test Passed
Completeness test
Successor Completeness Test Passed
Success: 1352, failure: 0, total expansions 611864, total generated 2477716
