MathDB

Three hoops are arranged concentrically as in the diagram. Each hoop is threaded with

20

beads,

10

of which are black and

10

are white. On each hoop the positions of the beads are labelled

1

through

20

as shown. We say there is a match at position

i

if all three beads at position

i

have the same color. We are free to slide beads around a hoop, not breaking the hoop. Show that it is always possible to move them into a configuration involving no less than

5

matches.

Problem Statement