A simple one this week….. how many squares are there on an 8×8 chessboard?

As ever, please do NOT post your answers, but do say whether you think you have solved it and how long it took.Β  Solution on Monday.

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. Nice!

      I wanted to say ‘A chess board features two squares. One plays as white and the other as black.’ but I quite like chess.

      I will say that I worked out the formula almost immediately, but am slightly drunk, so took almost all of my 2 minutes and 4 seconds doing simple arithmetic.

  1. Bit of a counting exercise, isn’t it? Figured out the route to the answer in a couple of seconds. More time spent checking I’d got them all.

  2. Thought of captain obvious answer for 2 seconds, then realised what it was asking. Thought I was in for lots of tedious counting, then noticed a pattern by the second set of squares. Grabbed calculator. ?????, Profit. Probably 2 mins.

  3. Worked out the pattern quite quickly so was able to work it out for a chessboard, glad the board wasn’t much bigger else I would have resorted to using a quick bit of code to do the work πŸ™‚

  4. 30 seconds “brute forcing” it, then spotted the correct way of solving it – one minute doing the mental arithmetic and double-checking.

    1. Dito…
      However presumably started with a short cut…
      However, the solution depends on how are the squares defined… if all squares are allowed that may be definded between all crossing points.. i would have to count again…

  5. I suppose we’re supposed to align the squares to the 8×8 small black and white squares and not break it down to the size of atoms or even space quantums (if any)? πŸ˜‰ Then I got the pattern pretty quickly and got the solution in, uhm, about a minute or a little more.

  6. 1 second to get the wrong answer, about 5 seconds to think of a method, about 45 seconds to add my numbers together in my head. About an hour more of work before I can look ata piece of paper ad confirm it!

  7. I knew immediately that the answer wasn’t the obvious one – a little playing around with a relevant sized grid, one sum, and I arrived at what is probably the answer Richard’s looking for.

    The nice thing is that you can easily scale up or down for boards of different sizes without needing to count (as long as you know the grid size!) πŸ™‚

  8. “Oh, that’s easy. There’s… wait… what about… so that’s… but they could also… I’m gonna need a piece of paper for this.”

  9. Very quickly because did this once in a maths class. Didn’t remember the answer but remembered n(n+1)(2n+1)/6! πŸ˜‰

    1. Challenge accepted πŸ™‚
      I summed up rectangles for half an hour and find an astonishing simple solution! In a grid with 8×8 squares (chessboard) the lines may outline 1296 different rectangles. The formula I found is

      Sum of n^3
      with n from 1 to 8
      = 1296

      I systematically summed up all the rectangles with the basis of 1, add those with 2 and so on…
      The formula for arithmetic series is helpful here. Please correct me if I’m wrong. I finally calculated a total of

      Sum of 2*n*n*n/2
      with n from 1 to 8

      which equals Sum of n^3.
      Now it’s not that hard to calculate the number of rectangles even for a 10×10 board…

  10. Simple formula, 30 seconds. However, most chess boards have a square border, so perhaps we should add 1 to the answer πŸ˜‰

  11. I’m sure there’s an elegant math solution… I set up a representation of a board in Excel and set up some formulas to calculate all the iterations. I think I got it right. Took about 5 minutes.

  12. Easy. There are 8 x 8 = 64 squares on a chess board. It is of course possible to draw a whole lot of new squares along the lines of the existing ones, but that is not the question here; it reads “how many squares ARE THERE on an 8Γ—8 chessboard” (my emphasis).

    1. I actually like this answer. Should it prove to be the intended answer on Monday I’ll have no problem claiming to have known it all along.

  13. about 10s to spot the pattern, not gonna bother to work out the number: I can wait until Monday and let Richatd do the number crunching.

    1. The answer does change depending on what kind of square you mean, doesn’t it.

  14. I knew it straight away, not because I’m smart (I’m not even close to being that) but because I took it in college.

  15. Caught the pattern quickly but had to prove to myself that it didn’t break down in actual application. Hey what can I say , I’m slow.

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