8th grade geometry. Day 1.

I am part of a wonderful network of teachers, brought together by Henri Picciotto, called Escape from the Textbook. I learned of this problem by one of the members of the group who uses it as a way to introduce the idea of formal proof. It’s what’s for breakfast tomorrow: 1st period of the 1st day of school. Fun stuff, I hope. After a bit of research on Wikipedia, I learned that this problem is called the Mutilated Chessboard Problem and was published in 1958 by George Gamow and Marvin Stern.

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Hi. You have a nice blog, and thank you for adding Math and Multimedia to your blogroll. If you want to promote your work, we would love to have one of your articles in the Math and Multimedia Carnival 15.

A carnival is a good place of promoting blogs.

Wow, I like it. You don’t have cutout chess board and domino piece documents to go with it by any chance do you? I think I could design some pretty quickly if not.

Chuck: I just gave student several copies of the chess board and they did fine without manipulatives (these were my 8th grade geometry students). They just outlined 1×2 rectangles and went through by trial and error. If you design something, please share it as I intend to use this problem with my 8th grade Pre-Algebra classes and for them I will definitely use manipulatives.

Did they all do it by trial and error? Or did you get anyone who came up with the argument that every domino has to cover both a dark and light square, and since there are two fewer dark squares (the ones that were x’d) it’s impossible to cover the entire board?

@Art: Yes, there were a variety of explanations that I would consider proofs. In my previous comment, I didn’t mean to imply that all they did was trial and error. They started there, and once they began to see that it never seemed to be possible, they worked together to try to go deeper. (with some questioning on my part…like look at the squares left over…is there anything in common for each trial?). My favorite student generated explanation of the proof went something like this: “So, if you have a school dance with 64 kids, half boys and half girls, and 2 girls leave to go to the bathroom, how can the remaining 62 kids pair up?”

If you’ve seen the path puzzles from James Tanton (the ones with boxes and you need to draw a line through all the boxes) you can throw that at them as well. It’s a similar reasoning for when it is possible to complete the path given a grid of certain dimensions and where the starting dot is placed.

I first saw your puzzle in Out of the Labyrinth by the Kaplans. Worth a read if you haven’t done so.

@jason: thanks for these additional puzzling resources…i’ll definitely check them out as I’m trying to bring in a lot of this kind of thinking into my 8th grade Pre-Algebra class (in addition to the geometry one).