Summary of MENTOR problem-solving session
Feb. 5, 2016

We consider three problems designed to improve your problem-solving skill set.

(1) On the rigidity of isometries

Often in mathematics, one structural assumption about the nature of an object will imply additional structural constraints. This happens, for example, in complex analysis. For complex-valued functions of one complex variable, the existence of the complex first derivative implies the existence of complex derivatives of all orders.

The problem: Prove that every isometry of Euclidean space that maps the origin to the origin must be linear.
Solution hint: Show that isometries preserve the inner product.

(2) A lattice question

The simple fact that the positive integers are well-ordered can be used in surprising ways.

The problem: Consider the square lattice in two-dimensional Euclidean space consisting of points with integer coordinates. Does there exist a regular hexagon such that all six of its vertices lie on the square lattice?
Solution hint: It is possible to solve this problem by arguing directly using trigonometry. The following hint points to an elegant solution that uses the fact that the positive integers are well-ordered. Suppose we have embedded a regular hexagon into the square lattice. Now rotate each of the six vertices 90 degrees about the next adjacent vertex. What happens?

(3) A counting problem

This problem appears on the 1996 Putnam exam.

The problem: Define a selfish set to be a set which has its own cardinality as an element. Find (with proof) the number of minimal selfish subsets of the first n positive integers. Here a minimal selfish set is a selfish set with no proper selfish subsets.
Solution hint: Argue inductively.