The Art and Craft of Problem Solving

54 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS

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Since both 3s + 7t and 5s + t are to be integers, (3s + 7t, 5s + t) is a lattice point. Con­

sequently, counting the ordered pairs (s,t) with 0 < s, t < 1 such that both 3s+ 7t and

5s + t are integers is equivalent to counting the lattice points inside the parallelogram.

This is easy to do by hand; the answer is 31. •

This problem can be generalized nicely using Pick's Theorem (Problem 2.3.38 on

page 52). See Problem 2.4.22 below.

Pictures Don't Help? Recast the Problem in Other Ways!

The powerful idea of converting a problem from words to pictures is just one aspect of

the fundamental peripheral vision strategy. Open your mind to other ways of reinter­

preting problems. One example that you have already encountered was Example 2.1. 7

on page 20, where what appeared to be a sequence of numbers was actually a sequence

of descriptions of numbers. Another example was the locker problem (Example 2.2.3

on page 29) in which a combinatorial question metamorphosed into a number theory

lemma. "Combinatorics � Number Theory" is one of the most popular and produc­

tive such "crossovers," but there are many other possibilities. Some of the most spec­

tacular advances in mathematics occur when someone discovers a new reformulation.

For example, Descartes' idea of recasting geometric questions in a numeric/algebraic

form led to the development of analytic geometry, which then led to calculus.

Our first example is a classic problem.

Example 2.4.2 Remove the two diagonally opposite comer squares of a chessboard.

Is it possible to tile this shape with thirty-one 2 x 1 "dominos"? (In other words, every

square is covered and no dominos overlap.)

Solution: At first, this seems like a geometric/combinatorial problem with many

cases and subcases. But it is really just a question about counting colors. The two

comers that were removed were both (without loss of generality) white, so the shape

we are interested in contains 32 black and 30 white squares. Yet any domino, once it