An Old Favorite

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The image above is found on the Nrich math site at


I first encountered this problem in 2014 in Jenks at a TMC session run by Megan (@Veganmathbeagle)

In the past two days I presented this to three of my classes – my Geometry class and my two Discrete Math classes. Much to my delight the classes all solved the problem and they all solved it different ways. In one Discrete class the group locked in right away on the fact that squares are worth two more than triangles. One of my students made a quick decision to attack this by a guess and check method and he, luckily, guessed correctly on the first try. We had a pretty good conversation about the strengths and weaknesses of relying on lucky guesses. In my second Discrete class there was a bit of debating about what clues to focus on. While they were tossing some good ideas around one student told us that none of our ideas mattered. Well, he was nicer than that but he did manage to circumvent all of our clever ideas by simply asking if he could add all the sums indicated in the right column and compare that to the sums indicated in the bottom row. It too a little convincing for his classmates to believe him, but they came around to his way of thinking. Interestingly (at least to me) some of the students still wanted to know the individual values of the shapes. In my Geometry class the students also focused on the difference between a square and a triangle. Before we went much further in that conversation, a student pointed out that the first and third columns only differed by a square turning into a triangle. Since we knew that squares were worth two more than triangles (again, they found this using the third and fourth rows) we can know that the question mark should be replaced by ______ (no spoilers here!)


I loved listening to the ideas bubbling out and I especially liked that they moved forward quickly in all three classes with nothing more than the visual prompt above. It’s great to hear the interactions and it is instructive to hear what they are focusing on when engaging with a problem like this. Fun problem solving in these classes. Later this week I intend to write about our Calculus exploits and revisit my ideas / frustrations with homework in my Geometry class.

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