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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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Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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.