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Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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All in the Mind

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

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Reflecting Squarely

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Concrete Wheel

Age 11 to 14 Challenge Level:

Alan Schoenfeld suggests that this problem tends to provoke immediate and widely divergent intuitive reactions (see some of the possibilities listed in the Hint section).

He writes that he used this problem in a class where the discussions "focused on what it means to have a compelling mathematical argument. The general tenor of these discussions followed the line of argumentation outlined in Mason, Burton and Stacey's (1982) Thinking Mathematically : First, convince yourself; then, convince a friend; finally, convince an enemy. (That is, first make a plausible case and then buttress it against all possible counterarguments.) In short, we focused on what it means to truly understand, justify, and communicate mathematical ideas."

Teachers may want to use this problem in their classrooms to serve the same purposes.