
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 define successor states for the 24 Game, we need to consider all possible ways to combine two numbers from the list using the allowed operations (addition, subtraction, multiplication, and division). Each combination will result in a new list of numbers, which represents a successor state.

Here's a step-by-step breakdown of how to generate successor states:

1. **Select Two Numbers**: Choose any two numbers from the list.
2. **Apply Operations**: Apply each of the four operations (addition, subtraction, multiplication, and division) to the selected numbers.
3. **Generate New List**: Replace the two selected numbers with the result of the operation, forming a new list of numbers.
4. **Repeat**: Repeat the process for all possible pairs of numbers and all operations.

For example, given the initial state `[6, 6, 2, 1]`:
- Select `6` and `6`, apply addition to get `12`, resulting in the new list `[12, 2, 1]`.
- Select `6` and `2`, apply subtraction to get `4`, resulting in the new list `[6, 4, 1]`.
- And so on for all pairs and operations.

Here is the Python function that returns a list of successor states for an input state:

python
```
def successor_states(state):
    successors = []
    n = len(state)
    
    # Iterate over all pairs of numbers in the state
    for i in range(n):
        for j in range(n):
            if i != j:
                a, b = state[i], state[j]
                
                # Generate successor states for each operation
                # Addition
                new_state = state[:]
                new_state.remove(a)
                new_state.remove(b)
                new_state.append(a + b)
                successors.append(new_state)
                
                # Subtraction (consider both a - b and b - a)
                new_state = state[:]
                new_state.remove(a)
                new_state.remove(b)
                new_state.append(a - b)
                successors.append(new_state)
                
                new_state = state[:]
                new_state.remove(a)
                new_state.remove(b)
                new_state.append(b - a)
                successors.append(new_state)
                
                # Multiplication
                new_state = state[:]
                new_state.remove(a)
                new_state.remove(b)
                new_state.append(a * b)
                successors.append(new_state)
                
                # Division (consider both a / b and b / a, if non-zero denominator)
                if b != 0:
                    new_state = state[:]
                    new_state.remove(a)
                    new_state.remove(b)
                    new_state.append(a / b)
                    successors.append(new_state)
                
                if a != 0:
                    new_state = state[:]
                    new_state.remove(a)
                    new_state.remove(b)
                    new_state.append(b / a)
                    successors.append(new_state)
    
    return successors

# Example usage:
initial_state = [6, 6, 2, 1]
print(successor_states(initial_state))
```

This function generates all possible successor states by applying each operation to every pair of numbers in the input state. The resulting lists of numbers are collected in the `successors` list and returned

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]

# Example usage:
goal_state = [24]
non_goal_state = [24, 1]

print(is_goal_state(goal_state))       # Output: True
print(is_goal_state(non_goal_state))   # Output: False
``

System message:
Goal Soundness Test Passed
Soundness test

System message:
Successor States Soundness Test Passed
Completeness test
Successor Completeness Test Passed
Success: 1352, failure: 0, total expansions 628081, total generated 5420036
