In this column we offer a selection of problems for solution.

Answering some of these questions will not require any particular mathematical knowledge. Others might require ‘prerequisite' to obtain a satisfactory answer. E-mail your solutions or submit your favorite puzzle if you wish to Jeno Lehel at jlehel@memphis.edu.

We will communicate the answers in our next Newsletter. Enjoy the problems!

ART GALLERY.

A hall in an art gallery has eight corners and eight straight walls from corner to corner. We would like to assign watchmen standing at appropriate corners in order to make sure that each wall is surveyed (see a solution in Figure 1 where two men are standing at corners A and B).

Figure 1

We do not know anything more about the shape of the octagonal hall. Can we always solve the problem with at most two watchmen?

BILLIARD

How to strike a ball on a circular billiard table in such a way that after hitting the cushion twice the ball will return to its original position? (Figure 2.)

CUT THE CAKE

A square cake (that is, its horizontal cross-sections are congruent squares) is frosted evenly on the four sides and the top. How can we cut the cake into five pieces (with vertical cuts) so that all the pieces have equal amounts of cake and equal amounts of frosting? (Figure 3.)

Figure 3

Can you cut the cake in the same way into p pieces, for every integer p 1 ?