What’s Your Problem?

Problems in the MathCounts School Handbook are presented “shotgun style,” that is, a geometry problem precedes a logic puzzle and follows a probability question. (I worked for MathCounts for seven years and then served as a writer and chair of their Question Writing Committee, so I’m not unbiased.)

By comparison, textbooks often present 50 exercises on the same topic, each one only minimally different from the previous one. That tips the hand to students, methinks, and makes them realize, “Oh, I just need to do the same thing.” I prefer the MathCounts approach, where students have to dig into their bag of tricks to find a viable solution strategy.

With that in mind, here are a few problems I’ve encountered recently, each one not like the others.

Problem 1. The simple polygon is made from 73 squares, connected at their sides. What is the perimeter of the figure?

A simple polygon made from 73 unit squares.

Problem 2. What is the expected number of times that a six-sided die must be rolled to get each number 1–6?

Problem 3. A wall is to be constructed from 2 x 1 bricks (that is, bricks that are twice as long in one direction as the other). A strong wall must have no fault lines; that is, it should have no horizontal or vertical lines that cut entirely through a configuration, dividing it into two pieces. What is the minimum size of a wall with no fault lines? The figure below shows a 3 × 4 wall that has both horizontal and vertical fault lines.

A 3 × 4 wall with a fault line.

Please share great problems you’ve recently encountered in the Comments.

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No answers, but here are some hints.

Problem 1. Look for a pattern.

Problem 2. Check out this simulation for the Cereal Box problem.

Problem 3. The smallest arrangement without a fault line is larger than 3 × 4 and smaller than 10 × 10.