### Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

### Coins on a Plate

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

### AP Rectangles

An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?

# Regular Hexagon Loops

##### Stage: 3 Challenge Level:

A hexagon loop is a closed chain of hexagons that meet along a whole edge and in which each hexagon must touch exactly two others.

They do not need to be symmetrical or short:

Can you find any rules connecting the numbers of tiles, the inside perimeter and the outside perimeter?

You might want to start by exploring square loops or growing sequences like these:

If you haven't got a supply of regular hexagons you could work on these regular hexagon sheets

With thanks to Don Steward, whose ideas formed the basis of this problem.