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How Many Squares on a Checkerboard?

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Now that I’m an official blogger (with two blogs posts under my belt), I found selecting the next problem to be a real dilemma. I have decided to post another “classic” problem.

How many squares are on a standard (8 x 8) checkerboard? 


As with the Handshake problem, the appeal of this problem (and what probably makes these problems classics) is its accessibility to students across many grade levels, the variety of problem-solving strategies that can be brought to bear in its solution, and the large number of variations/extensions. The simplicity in stating and setting up the problem is also part of its appeal.

A word of caution when introducing this task: Often students see this problem as somewhat trivial, counting just the 64 small squares; some go an extra step and realize that the whole board is also a square, for a total of 65. So, realize that students (or teachers) might need some prompting to recognize that the board also has 2 x 2 squares, 3 x 3 squares, and so on.

So, there you have it. Go ahead and have some fun with this task!

I was gratified to see the response to the launching of the TCM Blog. The site had lots of visits and a few comments. I’m hoping that for this post, we’ll get even more visits, and that more of you who visit the site will take the extra step to post a comment/question/random thought/whatever drawn either from your own experience/reflections or from introducing the problem in your classroom.

As with the first task, I’ll be back in a couple of weeks to post solutions/thoughts/extensions/variations to the task. I hope to hear from you soon and that you’re enjoying “Math Tasks to Talk About.”

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

I offered this problem to a group of 4th grade students who had not previously been exposed to problems like these requiring perseverance and creative, original thinking. Even though I added "how many squares of any size", they immediately told me 64, and then quickly corrected to 65 to include the whole board. It took some guidance to get them to see that there are many different size squares. But once they realized they could have 2x2, 3x3 etc. they tried to partition the board into even squares. It took some more work to have them see that the squares may indeed overlap, and even more guidance to convince them that they needed to find a systematic way of counting. Once they successfully counted the number of each size square, I asked if they noticed any patterns in the numbers. They are still considering this question so I hope to get back to you later. But I can report that the students enjoyed this new (for them) type of math activity and are excited about solving the next challenging task. Thanks for the great post!
Posted by: LisaE_35421 at 3/21/2014 11:06 AM

Thanks for the wonderful comment, Lisa! And what a great job you did with your students! You have effectively described the almost perfect "scaffolding" plan, and led your students through the main considerations for the problem! Will be interesting to hear what patterns they notice in the numbers!
Posted by: RalphC_79522 at 3/21/2014 1:37 PM

I agree, that the appeal of this problem is the fact that it is widely accessible to students across the grades. For younger students I think I would prefer to introduce a simpler problem--depending on the ability level a 2 x 2 or 3 x 3 grid. Older students could be challenged to revisit the 8 x 8 grid and find the number of rectangles on a standard checkerboard. Thank you for reminding me of this classic problem!
Posted by: ElizabethS_05318 at 3/27/2014 10:08 AM

Thanks for posting, Elizabeth. You're absolutely right that the place to start with younger students would be with a 2 x 2 grid, which just has them counting the individual squares and the whole square. Moving to the 3 x 3 grid is important, because this is where, besides the individual squares and the whole square, they have to visualize another "type" of square within (the 2 x 2 square), AND think about "overlapping" squares. Your suggestion of an extension for older students to consider the number of rectangles is an excellent idea (with the caution that it is a "far from trivial" task! :)
Posted by: RalphC_79522 at 3/28/2014 1:47 PM

My 3rd grade students have really stepped up with problem solving. Many were able to solve the sandwich measurement as well as the handshake problem but the 200 pound canoe problem really stumped them! It was great to see their imaginations at work; they could have been working on a writing prompt though not a math problem given their imaginative solutions!
Posted by: SteffanyC_83252 at 5/6/2014 11:07 PM

Offering a "simpler" problem could be a good way to use the math strategy of "finding" a simpler problem. It might be interesting to start with the whole checkerboard with young children and not assume they all need to begin with a simpler problem first. We might be surprised! After an initial go at the whole board, the teacher could then use the smaller partitioned squares as a way to both model the strategy of finding/making a simpler problem and as way to scaffold children who are truly engaged but don't have a clue where to begin, even after they have heard their peers' ideas.
Posted by: SaraD_38017 at 9/4/2014 1:31 PM

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