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The gap between two squares

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The question

Two square patches of floor are laid with the same tiles. The bigger patch takes 273 more tiles than the smaller, and both patches are between 14 and 25 tiles along each side.

How many tiles long is each patch?

Two squares apart, and the identity that makes it a list

gap:273· between14and25
a² − b²=(a + b)(a − b)
sum×difference=273
1×273→136and137
3×91→44and47
7×39→16and23
13×21→4and17
136and137
44and47
16and23
4and17
  1. The two squares are 273 apart, and both sides are between 14 and 25.
  2. Use the identity a² − b² = (a + b) × (a − b). It turns a difference of squares into a multiplication.
  3. So the two numbers' sum and their difference multiply to 273, and that is a list you can write down. Both must be even or both odd, or halving them back gives fractions.
  4. Each split gives a pair: half the sum plus half the difference, and half the sum minus it. That is 1 × 273 → 136 and 137; 3 × 91 → 44 and 47; 7 × 39 → 16 and 23; 13 × 21 → 4 and 17.
  5. The range decides it: only 16 and 23 are both between 14 and 25.

How it works.

  1. 01

    The two squares are 273 apart, and both sides are between 14 and 25.

  2. 02

    Use the identity a² − b² = (a + b) × (a − b). It turns a difference of squares into a multiplication.

  3. 03

    So the two numbers' sum and their difference multiply to 273, and that is a list you can write down. Both must be even or both odd, or halving them back gives fractions.

  4. 04

    Each split gives a pair: half the sum plus half the difference, and half the sum minus it. That is 1 × 273 → 136 and 137; 3 × 91 → 44 and 47; 7 × 39 → 16 and 23; 13 × 21 → 4 and 17.

  5. 05

    The range decides it: only 16 and 23 are both between 14 and 25.

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