A Maths Puzzle / How Clever is AI?

October 4, 2026

Solve for x, 3x = 2x

x = 0

On the face of it, the only possible solution to the equation is zero.
3 x 0 and 2 x 0 are both zero.
Any other value for x is impossible.

Using Algebra

In maths, algebraic equations require us to rearrange them so that x is on one side. One way to do this is to divide both sides by x. This leads to:

\frac{3}{2} = \frac{x}{x}

1.5 = \frac{x}{x}

Now, any number divided by itself is 1, but: 1.5 <> 1.

Can x be 0?

The trouble is if x was zero,

1.5 = \frac{x}{x}

The above cannot be defined because any division by zero is undefined in mathematics.
If you divide any number by zero on a calculator, it returns an error.

What does AI think?

To Consider

Has AI solved the puzzle algorithmically or using probability?
Has AI explained its solution by retrieving a stock answer or using probability?

1. Has AI solved the puzzle algorithmically or using probability?

Short answer: It solved it algorithmically, not probabilistically.

Why: The equation 3x = 2x is a deterministic algebra problem. There is only one mathematically valid manipulation:

$3 x - 2 x = 0 \Rightarrow x = 0$

This is pure symbolic algebra. There is no statistical inference, no pattern‑matching, no probability distribution involved.

Even though LLMs internally operate using probability, when the task is a well‑defined algebraic manipulation, the model follows the learned algorithmic pattern for solving linear equations:

isolate terms
subtract like terms
solve for the variable

This is not “guessing the next word” in a probabilistic sense; it is reproducing the correct algebraic procedure because that procedure is strongly encoded in its training.

So:

The puzzle itself is solved algorithmically.

2. Has AI explained its solution by retrieving a stock answer or using probability?

Short answer: It explained the solution using probability‑driven pattern generation, not by retrieving a stored answer.

Why: LLMs do not store fixed answers or templates. They don’t have a “stock response” for “3x = 2x”.

Instead, they generate each sentence by predicting the next token based on:

mathematical patterns learned during training
the structure of your question
the style you previously asked for
the conversational context

So the explanation is produced probabilistically — but guided by the strong mathematical patterns the model has learned.

This is why:

the algebra is correct
the explanation is tailored to your phrasing
the style matches the refined version you asked for yesterday

It’s not retrieval. It’s probabilistic generation constrained by learned mathematical rules.

So:

The explanation is generated probabilistically, not retrieved.

J K Mullins
The Book Hook Online
October 4, 2026