On Friday I set this puzzle….. how many squares are there on an 8×8 chessboard?

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

204!  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 (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.



  1. Yeap: you forgot the square on the back of the chessboard 🙂 That makes a real chessboard have 205
    (and even more if it has a border 😉 ).

    1. smallest squares: 8 x 8 on a board
      squares of 2 x 2 black/white combi: 7 x7 on a board
      etc, making it
      8² + 7² + 6² + … + 1 = 204
      + the one on the back = 205

    2. And really, if your chessboard is three-dimensional, and if you are going to be as technical as some people here, perhaps you should count the outside rim as one, also…

      so, 206! 😀

  2. Yes, got it.


    That’s all. Next week: how many rectangles are there on a chess board?

    1. As I posted on the puzzle-site from Friday:
      For Rectangles I found the following formula

      Sum of n³ for n from 1 to 8 = 8² * 9² / 4 = 1296

  3. Got it! After taking pencil and paper and outlining just a few of the different size squares and their possible locations, it was easy to see the pattern.

    For example, there are 8 possible locations across and 8 possible locations down for a square of 1 unit, making 8×8=64.

    Next, there are 7 possible locations across and 7 possible locations down for a square of 4 units, making 7×7=49.

    This goes on until, the last one being 1 possible location across and 1 possible locatin down for a square of 64 units, making 1×1=1.

    So 64+49+36+25+16+9+4+1=204

    1. Yes, thanks for posting the formula.

      Since the sum i for i from 1 to n = n(n+1)/2, it can be proven that the sum of i^2 for i from 1 to n = n^3/3 + n^2/2 + n/6
      ( n = 8, 170.67+32+1.33 = 204 )

      which can be simplified to n(n+1)(2n+1)/6 for easier substitution as you show. Thank You!

  4. Easy puzzle this week 🙂 Nothing out of the box required really.

    But hey, actually the real answer is infinity.

    You can fit an infinite number of squares on any board or square. The chess board is 8×8 units big, so surely a whole lot of squares the size of 0.001×0.001, 0.002×0.002, 0.003×0.003 etc would fit.

  5. hmm rectangles… lets see, by definition, squares are also rectangles.
    So, that would be 204 + 8*7*2 + 8*6*2 + 8*5*2 + 8*4*2 + 8*3*2 + 8*2*2 + 8*1*2 + 7*6*2+ 7*5*2 + 7*4*3 +……….

    the 8*x*2 series is for y*1 vertical rectangle + 2*1 horizontal rectangles. 7*x*2 is for y*2 rectangle, then you need to do y*4, y*5 etc.

  6. I used this Perl one-line program to calculate the 204:

    perl -MList::Util=sum -E ‘say sum map { $_ * $_ } 1 .. 8’

  7. well i got 204, it was quite straight forward. doing
    how many square that are:

  8. this is a batty man puzle.
    KMT. it was too hard bruv.
    ur a true numba jack.
    this aint even in english lyk. there is no mention of peng ne where.
    SNM. (in maths language QED.)

  9. I got 204, but unlike most posters above, I started with the one big square, then I visualized the 4 7×7 squares. When I got to the 9 6×6 squares I noticed that so far I had results of 1, 4 and 9 which I always remember as the dimensions of the monolith in 2001 A Space Oddesey and the first three perfect squares. I was kinda jazzed to see the pattern continue up to the 64 small squares. So anyway, I guess I did it backwards!

  10. No, I didn’t get it because I tried to just brute-force count them and lost the energy to care around 96. I’m smart enough to know the question intends that we include squares of different sizes and not just the 64 you can move on, but I’m not smart enough to write a formula to do it.

  11. This is another excellent mind boggling puzzle. Thanks again richard! i look forward to the next installment on friday. I appreciate your work son!
    hahahaha looking forward to meeting up for lunch one day.
    this is my website feel free to check it out.

  12. I got the idea, but failed to see the pattern. Of the explanations above, I found Jamie’s the easiest to follow.

  13. The correct answer is obviously 64.

    If I showed you a pattern of two white squares and two black ones, no one in his senses would call that pattern “a square”.

    204 is the correct anser two the question “How many different, but overlapping squares could you draw along the borders of the existing squares on a chess board?”

  14. I have only just looked at this puzzle and got the wrong answer. I then read the comments and realised wherer I was going wrong. I wonder if I would have realised on my own had a picture of a chess board been shown?

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