How Combinatorics Can Improve Logic Problem Solving in AI Models Like GPT-4
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How Combinatorics Can Improve Logic Problem Solving in AI Models Like GPT-4
Hello everyone, I wanted to share an observation that might bridge the gap between classical mathematics and AI reasoning. During my experiments with models like GPT-4 and Claude 3.5, I stumbled upon something intriguing: combinatorics—the branch of ma...
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How Combinatorics Can Improve Logic Problem Solving in AI Models Like GPT-4

Hello everyone,

I wanted to share an observation that might bridge the gap between classical mathematics and AI reasoning. During my experiments with models like GPT-4 and Claude 3.5, I stumbled upon something intriguing: combinatorics—the branch of mathematics dealing with combinations, permutations, and state spaces—can significantly enhance a language model’s ability to solve logic problems.

Let me explain how I arrived at this.

The Problem

I’ve been testing large language models with classic puzzles, such as the Wolf, Goat, and Cabbage problem:

A farmer must transport a wolf, goat, and cabbage across a river using a small boat. The boat can carry only one item at a time. If left alone, the wolf eats the goat, or the goat eats the cabbage. How does the farmer transport all items safely?

While seemingly simple, this problem poses significant challenges for many language models:

  • They often fail to track the state of the items (e.g., which bank each is on).

  • They struggle to reason through backtracking moves (e.g., taking the goat back to the starting bank).

The Breakthrough

To address this, I modified the way I described the problem. Instead of leaving it in plain English, I explicitly framed it as a combinatorial state space problem:

  1. Defined each state as a tuple (Farmer, Wolf, Goat, Cabbage), where each value is either Left or Right.

  2. Highlighted valid state transitions, considering the constraints (e.g., wolf and goat cannot be left alone).

  3. Clarified that the farmer could return to a prior state if needed.

When I presented the problem this way, other models (Claude Sonnet, O1-mini) solved it correctly. Structuring the problem with combinatorics made the steps explicit, eliminating ambiguity.

Why Combinatorics Works

Logic puzzles like this involve navigating state spaces, which is essentially a combinatorial problem:

  • There are ( 24 = 16 ) possible states (Farmer + 3 items, each on one of two banks).
  • Only a subset of these states is valid, due to constraints.
  • The problem boils down to finding a path through this state space.

By explicitly defining states and transitions, models can reason systematically rather than relying on inferred logic from plain English descriptions. This approach seems to align more naturally with how AI processes structured data.

Implications

While combinatorics is a well-known mathematical tool, I suspect its explicit integration into language model problem-solving has been underexplored. Here are a few takeaways:

  1. Structured Prompts Matter: Framing problems mathematically enables even less advanced models to perform complex reasoning.

  2. AI + Classical Math: Bridging natural language understanding with combinatorics (or similar tools) could significantly enhance AI’s logical capabilities.

  3. Potential Applications:

    • Teaching AI to reason through multi-step problems systematically.
    • Solving combinatorial optimization problems in logistics, planning, or game AI.

Questions for the Community

  1. Has anyone else experimented with combining combinatorics or formal mathematics with AI reasoning? If so, what were your results?
  2. Could this approach be extended to other challenging logic problems, like the Missionaries and Cannibals puzzle or Sudoku solving?
  3. How could tools like combinatorics be formally integrated into AI training or prompt engineering to improve reasoning?

I’m curious to hear your thoughts and any related experiments you’ve conducted!

Closing Thoughts

This discovery felt accidental—I wasn’t even familiar with the term "combinatorics" until GPT-4 mentioned it. But it has opened my eyes to how structured thinking can unlock AI’s potential. I hope sharing this here sparks new ideas and discussions.

Looking forward to your insights!

In order to do this, I have been collaborating with a custom GPT I made called Hexel, which has a custom prompt and RAG knowledge source that includes various pieces of information relating to the FORTH programming language and mathematics, among other things. I want to provide this information to someone who is both formally trained, and generally better at this than I am, so that they can carry it forward.

submitted by /u/petrus4
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