The same principle applies here.

With only three completed equations, there is not enough information to prove that one particular hidden rule is the only possible rule.

That’s why these puzzles are better understood as pattern-recognition challenges rather than formal mathematical proofs.

Why 45 Is Often Considered the Intended Answer

The 45 solution is particularly attractive because the same compact operation works on every line:

1 × (4 + 1) = 5

2 × (5 + 1) = 12

3 × (6 + 1) = 21

Therefore:

5 × (8 + 1) = 45

The rule is consistent from beginning to end and doesn’t require introducing the previous answer into the next equation.

It also creates a recognizable mathematical structure:

1 × 5 = 5
2 × 6 = 12
3 × 7 = 21
5 × 9 = 45

That consistency is one reason 45 is a strong candidate for the intended answer.

The Real Challenge Is Not Calculation

Notice something interesting about the puzzle.

None of the multiplication involved is difficult.

You don’t need advanced mathematics.

You don’t need algebra.

You don’t even need a calculator.

The difficult part is recognizing that the apparent operation isn’t ordinary addition.

That’s why these puzzles can feel frustrating.