108 Chapter 9

Iterate consequences. Apply an observation repeatedly. What happens when a pro-

cess is iterated? Pay attention to what changes and what is preserved to understand the

effects of iteration. Sometimes, even a simple idea can gain enormous power under

iteration.

Look for a better proof. The well-known chess advice is, if you see a good move,

look out for a better one. Similarly, in mathematics, when you have proved a theorem,

there might be a better proof lurking nearby. Search for it! Revisit your theorem and

proof later; sometimes a different context will enable you to see a streamlined or more

elegant argument.

Exercises

9.1 Which regular polygons can be found using vertices of the 2 × 1 brick tiling? Prove your

answer.

9.2 Which sizes of squares arise in the integer lattice? Does

√

5 arise? How about

√

17? Which

numbers arise exactly? Can you give a complete if-and-only-if characterization?

9.3 Prove that in the square lattice, any line segment joining two lattice points is the side of a

square whose vertices are lattice points.

9.4 Prove or refute the following: In the hexagonal lattice, any line segment joining two lattice

points is the side of a hexagon whose vertices are lattice points.

9.5 Prove that the square lattice is not necessarily invariant under the reflection swapping any two

lattice points.

9.6 Prove or refute the following: The hexagonal lattice is invariant under the reflection swapping

any two lattice points.

9.7 Can some nonsquare regular polygons arise from lattice points in some rectangular lattice, not

necessarily square? Exactly which regular polygons can arise in such a lattice?