×QvantCase
← All techniques

Squares · level 2

Squaring with reference numbers

A square is just a multiplication where both circles are equal, so the circle product is always a perfect square — the easiest possible last step.

Worked example: 97 × 97

reference number 100
97−3
97−3
cross-add: 97 −3 = 94
× reference: 9,400
circles: −3 × −3 = 9
= 9,409

Step by step

  1. 1

    Choose the reference number: 100

    97 is − 3 · 97 is − 3

    Write each number with its difference from the reference in a circle.

  2. 2

    Cross-subtract (or cross-add) diagonally

    97 − 3 = 94

    The other diagonal gives the same answer: 97 − 3 = 94.

  3. 3

    Multiply that result by the reference number

    94 × 100 = 9,400

  4. 4

    Multiply the two circled numbers

    (−3) × (−3) = 9

  5. 5

    Add it to the running total

    9,400 + 9 = 9,409

97 × 97 = 9,409

The theory behind it

Intuition

A square near a reference number is (n − d)(n + d) + d², so you can always trade one hard square for one easy product plus a small square.

Common pitfalls

  • ×Forgetting to add d² back.
  • ×Using a reference that is not actually close, which makes d² large and slow.

In the interview

Squares appear in variance, standard deviation and compounding two periods forward.