A square with three parts
Find 213² without multiplying 213 by itself. Split 213 into three parts: 200, 10 and 3. Switch on the crossings, the pieces where two different parts meet.
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- A square, side cut into three parts: a = 1, b = 2 and c = 3.
- Three squares sit on the diagonal: 1 + 4 + 9 = 14.
- Six crossings pair up: 2 + 3 + 6, twice: 22.
- All nine pieces add to 36.
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
The rule
Squaring a bracket of two terms gives four products. Squaring a side with three parts gives nine pieces. Here a, b and c are the three parts.
Three pieces lie on the diagonal, the line from corner to corner. There a part meets itself, so each piece is a square. Six pieces are crossings of two different parts. Every crossing has an equal twin across the diagonal. When the three parts have different lengths, the six crossings are rectangles.
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
In words: square each part. Then add twice the product of each pair of parts.
Two checks
With parts 1, 2 and 3: the squares are 1 + 4 + 9 = 14. The crossings are 2(2 + 6 + 3) = 22. Together 14 + 22 = 36. And (1 + 2 + 3)² = 6² = 36.
Now take c = 0. Every term with c becomes 0. What is left is a² + 2ab + b². That is the square of a bracket of two parts.
Using it
213 = 200 + 10 + 3. Take a = 200, b = 10 and c = 3. The squares are 40000 + 100 + 9 = 40109. The three products are ab = 2000, bc = 30 and ca = 600. The crossings are 2(2000 + 30 + 600) = 5260. So 213² = 40109 + 5260 = 45369.