Squares Near a Known Square: n±1 and n±2
Level 2 Mental squares · Reviewed by The NumThink editorial team · Last reviewed
Short version: (n + 1)² = n² + 2n + 1: 41² = 1,600 + 80 + 1 = 1,681. (n − 2)² = n² − 4n + 4.
This page is part of the NumThink Mental squares course. Read why the method works, watch it run on worked examples, then drill it in the practice box below until it feels automatic.
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No signup · instant feedbackWhy it works
(n ± 1)² = n² ± 2n + 1 and (n ± 2)² = n² ± 4n + 4 — expanding the square. So any square one or two steps from a square you know is a tiny correction away.
This is round-and-correct again — recognition finding a friendly neighbor (40², 50², 30²) and correcting outward.
The steps
- Find the nearest known square. 41² → 40² = 1,600.
- Correct by the formula. (40+1)² = 1,600 + 2×40 + 1 = 1,600 + 81 = 1,681.
- Downward works the same. 98² = 100² − 400 + 4 = 9,604.
Worked examples
Easy 31²
Show the thinking
30² = 900
900 + 60 + 1 = 961.
Answer: 961
Medium 41²
Show the thinking
40² = 1,600
1,600 + 80 + 1 = 1,681.
Answer: 1,681
Hard 52²
Show the thinking
50² = 2,500
(50+2)² = 2,500 + 4×50 + 4 = 2,500 + 200 + 4 = 2,704.
Answer: 2,704
When to use it
- Any number within 1–2 of a round square you know (20², 30², 40², 50², 100²).
- Below 100 too: 98² via 100² − 400 + 4 is the fastest route there is.
When NOT to use it
- Numbers ending in 5 — the ending-in-5 rule is even faster.
- Numbers near 50 — the 50-base method below is built exactly for that zone.
Common mistakes
Correction too small. 41² answered 1,600 + 41 = 1,641. Fix: The correction is 2n + 1 — the DOUBLE of the base plus one — not just n.
Forgetting the +4. (n+2)² answered as n² + 4n. Fix: The full expansion is n² + 4n + 4 — the trailing 4 is the fingerprint of a 2-step.
Check yourself: The correction moves the answer in the right direction: 41² must exceed 40² = 1,600 by about 80 — and 1,681 does.