How many squares are actually in an 8 by 8 chessboard?

Question

The internet is arguing about this again: How many squares are in a chessboard?

A chessboard is $8 \times 8$ . Most people say 64 (the $1 \times 1$ squares). But theres also:

$2 \times 2$ squares: $7 \times 7 = 49$
$3 \times 3$ squares: $6 \times 6 = 36$
...
$8 \times 8$ squares: $1 \times 1 = 1$

So total: $\sum_{k=1}^{8} k^2 = \frac{8 \times 9 \times 17}{6} = 204$

But wait, does this count ALL squares? What about "tilted" squares where the sides arent parallel to the board edges?

I saw a Reddit thread where someone counted tilted squares in a $4 \times 4$ grid by rotating coordinates. They claimed theres 16 tilted squares in a $4 \times 4$ board, making the total more than expected.

What is the total for an $8 \times 8$ including tilted squares? Is there a formula?

Alex Kim · Accepted Answer

For an 8 × 8 board, the total number of axis-aligned squares is: ∑ k = 1 8 k 2 = 8 ⋅ 9 ⋅ 17 6 = 204 For tilted squares (not parallel to board edges): A tilted square in an n × n grid is defined by two points: the center and one vertex. The number of distinct tilted squares of size s (where s is the side length in axis-aligned units after rotation) is ( n − s ) 2. For an 8 × 8 board, the total including tilted squares is: ∑ k = 1 7 ( n − k ) 2 k + ∑ k = 1 n ( n + 1 − k ) 2 = n ( n 2 − 1 ) ( n + 2 ) 6 For n = 8: 8 ( 64 − 1 ) ( 10 ) 6 = 8 ⋅ 63 ⋅ 10 6 = 840 So theres 840 total squares (axis-aligned + tilted) on an 8 × 8 chessboard! This includes squares with sides at any angle — not just 45 degrees.

Alex Kim · Answer

The compact formula is: 1 2 + 2 2 + 3 2 + ⋯ + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 . For an 8 × 8 board: 1 2 + 2 2 + ⋯ + 8 2 = 8 ⋅ 9 ⋅ 17 6 = 204. Why does this work? A k × k square can start in 9 − k horizontal positions and 9 − k vertical positions, so there are ( 9 − k ) 2 of them. As k goes from 1 to 8, that gives 8 2 + 7 2 + ⋯ + 1 2 = 204.

How many squares are actually in an 8 by 8 chessboard?

2 Answers

How many squares are actually in an 8 by 8 chessboard?

2 Answers