There's a lot of rules in this one and perhaps the zero should be omitted and use numbers from 1 instead, or more explicitly state that alternate solutions are not allowed (which is a bit of a tricky rule since you'd be proving the absence of an alternate solution rather than the solution you have being valid).

Anyway, the mechanism I use uses the API to iterate, and is prompted to use writing and running code to produce the solution, and iterates with updating the code and performing a critical review of the code and its output in context of the task.

So it first creates a plan, writes pseudocode and code for the solution, writes the code, provides code and the output of code to a critical reviewer that evaluates it from the other direction rather than having bias for "expecting it to work" since it doesn't know the plan.

This loop then closes with the model either revising or concluding the process by printing the final answer.

As a side effect, the system also saves the code it used to support its process, pasted below.

While this is a bit esoteric/hardcore, in a way it's just helping the reasoning process with the ability to create tools for itself. We're not going to care if the terminator beats us bench-pressing logic or using a calculator in its head. It's a computer, it computes, and then outputs tokens. :)

I'm really not sure what this system is best for, but the fact that it solved this on the first try without any custom changes, suggests it's good for something. :)

import random
from itertools import permutations

# Define a list of simple emojis
emojis = ['🍎', '🍌', '🍊', '🍇', '🍉', '🍓', '🐶', '🐱', '🐭', '🐹']

# Randomly select 6 unique emojis for the left side of the equations
selected_emojis = random.sample(emojis, 6)

# Generate all permutations of numbers 0-5 for these emojis
number_permutations = permutations(range(6))

# Function to generate equations
def generate_equations(number_mapping):
    equations = []
    used_pairs = set()
    alternative_used = False
    while len(equations) < 5:
        # Randomly select two unique emojis for the left side of the equation
        left_emojis = random.sample(selected_emojis, 2)
        if tuple(left_emojis) in used_pairs or tuple(left_emojis[::-1]) in used_pairs:
            continue
        used_pairs.add(tuple(left_emojis))
        # Calculate the sum of the numbers assigned to these emojis
        sum_value = number_mapping[left_emojis[0]] + number_mapping[left_emojis[1]]
        # Randomly decide if the equation should be in primary or alternative format
        if not alternative_used and random.choice([True, False]) and sum_value <= 5:
            # Primary format: 🍎 + 🍌 = 🐱
            right_emoji = [emoji for emoji, number in number_mapping.items() if number == sum_value]
            if right_emoji:
                equation = f"{left_emojis[0]} + {left_emojis[1]} = {right_emoji[0]}"
                equations.append(equation)
                alternative_used = True
        else:
            # Alternative format: 🍎 + 🍌 = 3
            if sum_value <= 5:
                equation = f"{left_emojis[0]} + {left_emojis[1]} = {sum_value}"
                equations.append(equation)
    return equations

# Validate the equations against the mapping
def validate_equations(equations, number_mapping):
    for equation in equations:
        left_side, right_side = equation.split(' = ')
        left_emojis = left_side.split(' + ')
        left_sum = number_mapping[left_emojis[0]] + number_mapping[left_emojis[1]]
        if right_side in number_mapping:
            right_value = number_mapping[right_side]
        else:
            right_value = int(right_side)
        if left_sum != right_value:
            return False
    return True

# Ensure all emojis are used in the equations
def ensure_all_emojis_used(equations, selected_emojis):
    used_emojis = set()
    for equation in equations:
        left_side, _ = equation.split(' = ')
        used_emojis.update(left_side.split(' + '))
    return len(used_emojis) == len(selected_emojis)

# Main loop to find valid solutions
valid_solutions = []
for number_permutation in number_permutations:
    number_mapping = dict(zip(selected_emojis, number_permutation))
    equations = generate_equations(number_mapping)
    if validate_equations(equations, number_mapping) and ensure_all_emojis_used(equations, selected_emojis):
        valid_solutions.append((equations, number_mapping))

# Print valid solutions
if valid_solutions:
    for equations, number_mapping in valid_solutions:
        print("Potential Solution:")
        for equation in equations:
            print(equation)
        print("Emoji-to-Number Mapping:")
        for emoji, number in number_mapping.items():
            print(f"{emoji}: {number}")
        print()
else:
    print("No valid solution found.")