Difference of Squares

Technique: Spot the round midpoint

When two factors are symmetric around a round midpoint, the product collapses: (m−d)(m+d) = m² − d². Square the midpoint, subtract the offset squared.

Examples

• 18 × 22 → midpoint 20, offset 2 → 400 − 4 = 396
• 47 × 53 → midpoint 50, offset 3 → 2500 − 9 = 2491
• 14 × 16 → midpoint 15, offset 1 → 225 − 1 = 224
• 32 × 48 → midpoint 40, offset 8 → 1600 − 64 = 1536

How to spot it

Add the two factors. If the sum is even, the midpoint is just (sum)/2. The offset is half the difference. In this round the midpoint is always a multiple of 5 (so squaring it is fast — multiples of 10 just append 00, multiples of 5 use the ending-in-5 trick: n5² = n(n+1) | 25).

Why does this work?

Multiply out: (m−d)(m+d) = m² + md − md − d² = m² − d². The cross-terms cancel exactly. This is the same identity as Squaring Spree (R5), but applied to products of two distinct numbers rather than to a single square.

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