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Fast Squaring

Two rules cover almost every square you will meet: one for numbers ending in 5, one for numbers sitting near a round value.

8 min readLevel ★★Prerequisite: LR multiplication

1. The two rules

  1. Ends in 5: take the front part f, compute f × (f+1), then append 25. Example: 65 → 6 × 7 = 42 → 4225.
  2. Near a round number: write n = r + k (or r − k) with r round, then n² = r² + 2rk + k². Example: 97 = 100 − 3 → 10000 − 600 + 9 = 9409.
  3. The symmetric version: n² = (n−k)(n+k) + k². Example: 97² = 94 × 100 + 9 = 9409.

Which one to use

Last digit 5 → use rule 1, it is instant. Otherwise find the nearest round number: distance 1–2 → rule 3, distance 3 or more → rule 2 (r² is already memorised).

2. Worked examples

  • 35² → 3 × 4 = 12 → 1225
  • 145² → 14 × 15 = 210 → 21025
  • 62² = (60 + 2)² = 3600 + 240 + 4 = 3844
  • 97² = (94)(100) + 9 = 9409

Check the last one with DS: 97 → 16 → 7, and 7 × 7 = 49 → 4; DS(9409) = 22 → 4 ✓

3. Live demo

Type a number and let the site pick the fastest route

The demo is computed in your browser; nothing is sent anywhere.

4. Practice

  1. 35²
  2. 85²
  3. 145²
  4. 97²
  5. 62²
Answers
1) 1225.   2) 7225.   3) 21025.   4) 9409.   5) 3844.
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