May 12, 2026
How many squares are actually in this 8 by 8 chessboard puzzle viral problem explained
The internet is arguing about this again: How many squares are in a chessboard?
A chessboard is . Most people say 64 (the squares). But theres also:
- squares:
- squares:
- ...
- squares:
So total:
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 grid by rotating coordinates. They claimed theres 16 tilted squares in a board, making the total more than expected.
What is the total for an including tilted squares? Is there a formula?
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1 Answer
For an $8 \times 8$ board, the total number of axis-aligned squares is:
$$\sum_{k=1}^8 k^2 = \frac{8 \cdot 9 \cdot 17}{6} = 204$$
**For tilted squares (not parallel to board edges):**
A tilted square in an $n \times n$ 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 $8 \times 8$ board, the total including tilted squares is:
$$\sum_{k=1}^7 (n-k)^2 k + \sum_{k=1}^n (n+1-k)^2 = \frac{n(n^2-1)(n+2)}{6}$$
For $n=8$: $\frac{8(64-1)(10)}{6} = \frac{8 \cdot 63 \cdot 10}{6} = 840$
So theres **840 total squares** (axis-aligned + tilted) on an $8 \times 8$ chessboard!
This includes squares with sides at any angle — not just 45 degrees.
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