64-204 Squares on an 8×8 chessboard (The Game – The Math)

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There are a total of 64 individual squares on an 8×8 chessboard. When considering groups of squares, creating larger squares, there are actually 204 different squares that can be created on an 8×8 chessboard, including the largest 8×8 single square.

In this short article, I will explore how this is possible and show you some examples.

The Game

The standard chess board is a square in itself divided into 8 rows of 8 squares, therefore in terms of a game of chess there are 64 squares in total on a chess board that can be utilized throughout the game.

32 of these are dark squares and 32 are light squares.

At the beginning of the game, half of the squares are occupied by the 32 chess pieces in a set, therefore the maximum number of squares that can be used at any one time is 32, which will diminish as pieces are captured by the enemy and removed from the board.

The Math

Often the question of how many squares on a chessboard incorporates not only the usable squares we see on the board, but also how they can be combined into groups, to create larger squares from 2×2 squares right through to the singular 8×8 square which when totaled together add up to 204 squares.

To work out how many squares can be potentially created, we need to look at how many 2×2 squares there are on a board, then how many 3×3 squares are on the chess board and so on.

The answer to each of those questions are below, and the sum of each of these groups is then ascertained to be 204 possible combinations of different sized squares in total on a chess board.

• 1 x 8×8
• 4 x 7×7
• 9 x 6×6
• 16 x 5×5
• 25 x 4×4
• 36 x 3×3
• 49 x 2×2
• 64 x 1×1

Therefore, 1+4+9+16+25+36+49+64=204

I have seen many different ways for this to be explained both visually and in text but by far the best example is this animated explanation from datagentics.

What should the answer to how many squares are there on a chess board be if it is a question in a quiz?

The safest way to answer this question should it come up in a quiz, is to provide both answers of 64 and 204 and be prepared for the quizmaster to provide both answers, or just one. At which point, most contestants will claim a point for the 64 answer, whilst many will not have the 204 should that be provided.

Armed with everything you have learned in this article, you are going to be in a good position to explain yourself for having both answers.

And you never know, the question might be asked by the quiz master followed by the fact they will give a point for each answer – now you are guaranteed 2 points and will probably gain an advantage over the other teams in the quiz.

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