
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 successor states, we need to take every pair of numbers from the list, apply one of the four operations (addition, subtraction, multiplication, division), and generate a new state with the result of that operation and the remaining numbers. Here's how to implement this:

1. Iterate over all pairs of numbers in the list.
2. For each pair, perform all valid operations.
3. Generate new states with the result of the operation and the remaining numbers.
4. Return the list of all possible new states.

Here's the function:

```python
from itertools import permutations

def generate_successor_states(state):
    def valid_operations(a, b):
        ops = [(a + b), (a - b), (b - a), (a * b)]
        if b != 0:
            ops.append(a / b)
        if a != 0:
            ops.append(b / a)
        return ops

    def generate_new_states(state):
        n = len(state)
        new_states = []

        for i in range(n):
            for j in range(i + 1, n):
                for result in valid_operations(state[i], state[j]):
                    new_state = [state[k] for k in range(n) if k != i and k != j] + [result]
                    new_states.append(new_state)

        return new_states

    return generate_new_states(state)
```

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 len(state) == 1 and state[0] == 24
```

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 627424, total generated 2706100
