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← The Arithmetic of the Chessboard

The Arithmetic of the Chessboard

Three short lessons on the numbers in sixty-four squares: coordinates, the old doubling story, and how many squares there really are.

Spark & Whetstone · sparkwhetstone.org/learn/the-arithmetic-of-the-chessboard · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Coordinates

Every square has a name: a letter for the file (a to h, left to right from White’s side) and a number for the rank (1 to 8, from White’s side up). The white king starts on e1.

That is the same idea as a map grid or a graph: two numbers, one across and one up, pick out exactly one place.

Question. A knight moves two squares in one direction and one at right angles. From d4, which of these can it reach?

  1. d6
  2. f5
  3. e5

Say it back. In your own words: how do two coordinates pick out one square?

Lesson 2. The doubling story

An old story tells of a reward of one grain of wheat on the first square, two on the second, four on the third, doubling every square. The totals run 1, 3, 7, 15, 31: always one less than the next square’s amount.

So all sixty-four squares hold 2⁶⁴ − 1 grains, about eighteen billion billion, far more wheat than the world grows in a year.

Question. How many grains are on the first ten squares together?

  1. 1,023
  2. 1,024
  3. 512

Say it back. In your own words: why is the total always one less than the next square?

Lesson 3. How many squares

A chessboard has 64 small squares, but count every square of every size and there are many more: 1 × 1, 2 × 2, all the way up to the whole board.

A k × k square fits in (9 − k) positions across and (9 − k) down. Adding up 8² + 7² + … + 1² gives 204.

Question. On a 4 × 4 board, how many squares are there of every size?

  1. 30
  2. 16
  3. 20

Say it back. In your own words: why are there more than 64 squares on a chessboard?

At the table: Rice on the board

Players: Two or more, ages 7 and up. You'll need: A chessboard (or a drawn 8 × 8 grid), a cup of rice.

  1. Put 1 grain on the first square, 2 on the next, 4, then 8, doubling.
  2. Before going on, everyone guesses which square will first need more than the whole cup.
  3. Keep going until the rice runs out, and count how far you got.
  4. Work out roughly how many cups the last square would need.

Answers

  1. f5 — d4 to f5 is two files right and one rank up, a knight’s move. e5 is a diagonal step and d6 a straight one.
  2. 1,023 — Square 11 holds 2¹⁰ = 1,024 grains, so the first ten together hold 1,024 − 1 = 1,023.
  3. 30 — 16 + 9 + 4 + 1 = 30.