A cube of a sum
A stack of bricks is 2 bricks along each edge. Add a layer b bricks thick to three faces at one corner. Here b = 1. How many bricks are in the new cube? Slide b.
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- A cube, drawn as two flat layers: bottom, height a; top, height b.
- None new: a³ = 8.
- One new, three blocks: 3a²b = 12.
- Two new, three blocks: 3ab² = 6.
- All new: b³ = 1.
- Total: 8 + 12 + 6 + 1 = 27. So (a + b)³ = a³ + 3a²b + 3ab² + b³.
The rule
The new cube is 3 bricks along each edge. So it holds 3 × 3 × 3 = 27 bricks.
Cut the new cube into blocks. A block is a box-shaped group of bricks. A block has a length, a width and a height. Each one is a or b. A direction is new when it is b.
Sort the blocks by how many directions are new. One block has none new. One block has all three new. The one new direction can be any of three. So three blocks have exactly one new direction. The same holds for two new directions.
(a + b)³ = a³ + 3a²b + 3ab² + b³
At a = 2 and b = 1 the sum is 8 + 12 + 6 + 1 = 27. That matches 3³ = 27.
Each 3 counts the three choices of direction.
Putting −b in place of b
(a − b)³ = a³ − 3a²b + 3ab² − b³. Signs alternate. Odd powers of −b are negative. Even powers are positive.
| Old cube edge | New cube edge | Four kinds added | Sum |
|---|---|---|---|
| 2 | 3 | 8 + 12 + 6 + 1 | 27 |
| 3 | 4 | 27 + 27 + 9 + 1 | 64 |
Watch out!
Cubing each part and adding gives a³ + b³ = 9. The real cube holds 27. Cubing the parts alone leaves out six blocks, which hold 18 bricks.