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The gap between two squares, step by step

Two square gardens sit side by side.

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The question

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. One number is given and it is not either of the ones wanted: the two squares are 273 apart, and both sides are between 14 and 25. Squaring every number in that range and subtracting in pairs would work and is not a method.
  2. The way in is one identity: a² − b² is always (a + b) × (a − b). It is usually met the other way round, for factoring — here it is doing the opposite job, turning 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 have to be even or both odd — they differ by twice the smaller number, which is even — 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. Now the second fact does the choosing: only 16 and 23 are both between 14 and 25. Every other pair really does have squares 273 apart, which is why the range is not decoration — it is half the question.

How it works.

  1. 01

    One number is given and it is not either of the ones wanted: the two squares are 273 apart, and both sides are between 14 and 25. Squaring every number in that range and subtracting in pairs would work and is not a method.

  2. 02

    The way in is one identity: a² − b² is always (a + b) × (a − b). It is usually met the other way round, for factoring — here it is doing the opposite job, turning 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 have to be even or both odd — they differ by twice the smaller number, which is even — 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

    Now the second fact does the choosing: only 16 and 23 are both between 14 and 25. Every other pair really does have squares 273 apart, which is why the range is not decoration — it is half the question.

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