A favorite problem-solving problem among middle school and high school mathematics teachers is “How many squares are on a chessboard?” There is a nice twist for understanding (squares can be different sizes), some room for technique differences (counting by drawing, by reasoning, by laying a cut-out over actual squares, etc), and two major plans that work: an organized count, working with different-sized squares, and a solution by a pattern hunt: solve the problem for a 1×1 chessboard, then 2×2, then 3×3, and so on.

[update:  Image (and nice solution) from Nigel.]

I have a third (lovely) solution: For each little square on the board, count how many squares it is the northwest corner of. So, each of 15 squares on the south or east edge is the nw corner of only one square. The 13 adjacent squares are the nw corner of two squares each… We end up with 15×1 + 13×2 + 11×3 + 9×4 + 7×5 + 5×6 + 3×7 + 1×8…. This is nice with a fast class for use during the final “looking back” phase. Can the kids explain the relationships between the three methods of solution.

Anyway, today’s problem is, imo, richer. Use the space below for questions/comments, and click here to share various answers. There is wider variety of solutions with today’s problem than with the squares, and more room for discussion up front.

1. May 18, 2007 pm31 2:12 pm 2:12 pm

i’ve noticed that this page hotlinks an image on my page entitled how many squares / rectangles are there on a chess board???

seems only polite to link back:
http://puzzles.nigelcoldwell.co.uk/twentyseven.htm
I think i explained the solution quite nicely. ;)

2. May 18, 2007 pm31 2:39 pm 2:39 pm

Yes, you did. I should check my linking more carefully – I thought when I grabbed the image I had created a link…
Anyhow, I updated the post as well. Credit where credit is due!

June 13, 2007 pm30 3:26 pm 3:26 pm

YES IT IS VERY GOOD SITE

June 13, 2007 pm30 3:27 pm 3:27 pm

THIS IS VERY NICE PUZZLE SITE

August 30, 2009 pm31 4:02 pm 4:02 pm

If you are too lazy to look at the website or have a slow computer the answer is 1296. THE WEBSITE IS VERY SIMPLE BUT EASY. IT IS WORTH A LOOK.

• September 6, 2009 pm30 12:38 pm 12:38 pm

November 7, 2009 am30 10:51 am 10:51 am

but i want a rule to count it
please because i have a project on this
if any body know please send me an email on (romiovalantino@ymail.com)
i am a girl my name is alina and i cn do any thing for the one who know and send me an email
ALINA

November 7, 2009 am30 10:55 am 10:55 am

hello my name is alina
iam a girl
i want a rule for how to count a chess board square.
please if any body know please send me an email on (romiovalantino@ymail.com)
please i can do any thing for the one who will send me this email

• November 8, 2009 am30 8:54 am 8:54 am

For counting squares, it’s not so hard. Try this:

Draw a square. How many squares is it? 1. Easy
Now draw a 2×2 square. How many squares is it? 4 little ones, and the whole thing. 5.
Now try a 3×3, like a tic tac toe board with the edges included. Do you see all 14 squares? (Hint, there are 3 different sizes).

Keep working your way up. You’ll get to 8×8.

March 12, 2010 am31 3:55 am 3:55 am

total number of squares are 204

July 10, 2010 pm31 9:56 pm 9:56 pm

In that picture
there are 33 should you include the jagged outer one.

When they are glued together there are 64 (+ the outer)

When Engraved 64 (+the outer)

Really should you decide to encapsulate each group to produce
square results of linear tiles…. the result would end up cube/Pi

Then whats the point. There is no molecular density to compare
No sound wave, or magnetic emf

For a second grader to count as many squares as they can without
duplication. Is to kill time and keep the students out of the
instructors hair.

The problem is a farce, redundant and worthless.

Teach something useful Like Trig, how about what angle does the
light have to be positioned to create exactly a 3 inch shadow from

a 4 inch chess peice….. That you can use later in life.

8. November 12, 2009 pm30 5:37 pm 5:37 pm

Hi,

May 7, 2010 am31 6:28 am 6:28 am

hello To All Members.Wish you all guys wishes be ok.

May 30, 2010 am31 6:14 am 6:14 am

hello to every one.Wish you all whim be ok.

June 6, 2010 am30 3:38 am 3:38 am

Where r u guys studying rite now?

12. July 11, 2010 pm31 6:46 pm 6:46 pm

Just drawn back here by the new comment – I missed this one first time round.

I actually think rectangles is easier than squares (at least for me). Imagine you cut your chosen rectangle out with a guillotine. There are nine vertical lines on the board; you need one to start the rectangle and one to finish it, so that’s 9C2 = 36 possibilities. Same vertically, so in all you’ve got 36^2 rectangles.

I can’t think of anything that nice for the squares problem; I can justify (10C3 + 9C3) or (2*9C3 + 9C2) by counting arguments, but they’re less sweet. (The latter is nicer. The 9C2 counts squares symmetrical about the main diagonal, by the “edges” argument above. One of the 9C3s counts squares left of the main diagonal. Number the horizontal lines 0-8, the verticals similarly, and you’ve got three different numbers in order. The smallest is the left edge of the square, the second-smallest the top edge, the third smallest the bottom edge. The other 9C3 is symmetrical with this.)

August 22, 2010 pm31 10:29 pm 10:29 pm

Hi

We can count the number of squares in following manner.
Let there be n*n grid. Then the number of 1*1 squares is n*n. The number of 2*2 squares is (n-1)*(n-1). The number of 3*3 squares is (n-2)*(n-2). Continuing the same way the number of (n-1)*(n-1) squares will be 2*2. The number of n*n squares is 1*1. So now we can add the number of squares in n*n grid as 1^2 + 2^2 + 3^2 + 4^2 + ……. + n^2.

Vishal

13. March 15, 2011 am31 11:25 am 11:25 am

How many rectangles? Wow, it would be hard enough to count the squares, but the rectangles too?