On Friday I posted this puzzle….

How many squares are there on an 8 x 8 chessboard?

If you have not tried to solve it, have a go now. For everyone else the answer is after the break.

There are 8 x 8 individual squares, then 7 x 7 squares made up of 4 squares, then 6 x 6 squares made up of 9 squares, and so on, giving a total of 204 squares. Did you solve it?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called **PUZZLED** and is available for the **Kindle** (UKhere and USA here) and on the **iBookstore** (UK here in the USA here). You can try 101 of the puzzles for free here.

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Oh good, that’s what I got on Friday… then someone posted that answer (which they shouldn’t have done) and someone said it was wrong. So I started to doubt myself but I couldn’t see how I was wrong. (No doubt someone will challenge the answer in the next few hours.)

It occurs to me now that the person who said ‘Wrong’ on Friday might have been talking about the fact that someone had posted the answer, as in ‘That was wrong to do that.’ And not commenting on the actual answer itself. Still, who cares? I got one right for once.

Merry Christmas to one and all.

If you ever want to figure out how many squares are in a large square that has been subdivided into equal squares (like a checkerboard – or chessboard for the more cerebral):

Start with the biggest size.

Count down to 1, by 1s.

For each level, the number of that size squares will be the square of the level of the progression.

So, for this example

Number of 8×8 squares = 1 squared = 1

Number of 7×7 squares = 2 squared = 4

Number of 6×6 squares = 3 squared = 9

Number of 5×5 squares = 4 squared = 16

Number of 4×4 squares = 5 squared = 25

Number of 3×3 squares = 6 squared = 36

Number of 2×2 squares = 7 squared = 49

Number of 1×1 squares = 8 squared = 64

Add up the numbers on the right (1+4+9+16+25+36+49+64) and you get 204 squares.

thanks. That’s a nice, simple method I can remember. (The formula some others have posted looks simpler, but there’s no way I’d be able to remember it.)

Sorry for a late reply. But indeed it was both wrong to post an answer, and IMHO the answer is in fact wrong. A Chessboard is a 3D thing, so even ignoring the sides (that usually aren’t squares but rectangles), the backside of a chessboard also forms a square, giving a total of 205 squares.

Also most real-life chessboards have an outer edge (where sometimes the letters and numbers are placed), thus forming an even bigger square on the front, giving a total of 206 squares on most real-life chess-boards or even better 207, as the edge also makes up for a second square on the back.

Apparently Richard doesn’t own a real-life chessboard, but only an imaginary one that has no edges and no backside :-)

Pedants of the world unite!

Consider size of one side of chessboard (8). Double and add 1 (17). Multiply that by size (17 x 8). Multiply that by one more than the size (17 x 8 x 9). Finally divide by six. Works for any size.

There are only 64 marked squares. Are you inventing more ?

I do not share your opinion.. for me there are only 32 black squares on a white background… If the background has alos the shape of a square there would be 33 squared… ;-)

That’s obviously wrong Berhard. Surely there are 32 WHITE squares on a BLACK background!

Are you so short minded, Mad Kev?

Compared to last week that was a poor attempt at a puzzle.

Here’s the formula. Given a board n x n, the number of squares are (n(n+1)(2n+1))/6

Dear Richard,

With best wishes for Christmas and the New Year,

Hamish

The correct answer is zero. The only things “on” a chessboard are chess pieces! :-)

Square is an abstract anyway, so if you want to nitpick, I think you should say infinite.

For example there will be a square with length=1 cm in the air resting on the board. There will also be a square with length 1.1 cm, 1.11 cm, 1.111cm and so on.

Merry christmas!

I think that abstracts the puzzle to meaninglessness. Your answer only makes philosophy majors feel stupid for not thinking of it, while my answer makes all liberal arts majors feel stupid.

I think it’s more proper to say that the square of each space represent 64 real squares, and 140 “virtual” squares are derived from them. But, by that approach, we also have many other virtual squares that are defined by the intersection points on the board, starting with the 4×4 square that connects the side centers diagonally. By my count, there are 150 such additional squares, bringing the total to 354.

Now we have an answer that makes even the science majors look stupid. It truly *is* a merry day indeed! :-)

Just to head the pedants off at the pass: I meant 4×4 from a notational standpoint only. I know the length of the sides of the diagonal squares should be multiplied by the sqrt(2).

Merry Christmas, Richard, and thanks for the year round entertainment, though I know we’re really your psychology subjects.

Eddie

The answer to how many squares are on an n by n checkered board is the nth pyramidal number (2n^3+3n^2+n)/6, which is the closed form for the sum as i goes from one to n of i squared.

I ended up with a weird script essentially calculating the number of every square size 1-8 on the checkerboard.

This is done by calculating the number of squares touching the outer rim on concentric checkerboards of smaller and smaller sizes, until the square in question can’t fit on the checkerboard anymore. I suppose it could be describe as a series of sums, not the (2n^3+3n^2+n)/6 like others did.

I always like riddles, where scripts can help convince me about the answer :)

Merry christmas!

How about trying to play on “these” squares, your opponent would throw you out.

For me it’s 205 because you have to add the chess board itself!!!