### Icosagram

Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at the number of diagonals?

### Playground Snapshot

The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

### LOGO Challenge - Following On

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

# Heptagon Has

##### Stage: 3 Short Challenge Level:

It can possess all five properties.
The description does not say that heptagon has to be convex, i.e. all of its interior angles need not be less than $180^{\circ}$. Since $(2 \timesĀ 7 - 4) \times 90 = 900$, the interior angles of all heptagons total $900^{\circ}$. The creation of a heptagon with all the given conditions is possible as the diagram shows. Notice that four of the interior angles are acute and the other three are reflex angles.

This problem is taken from the UKMT Mathematical Challenges.