### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

### Pick's Theorem

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

### Producing an Integer

Multiply a sequence of n terms together. Can you work out when this product is equal to an integer?

# Super Computer

##### Age 14 to 16 ShortChallenge Level

$6^1 = 6$
$6^2 = 36$
$6^3 = 216$
$6^4 = ...6$ since the last digit comes from multipying the previous last digit by $6$

Last digit of $66^{66}$ is $6$

Divide by $2$: Last digit will be $3$ or $8$

$66^{66}$ will be a multiple of $4$ (because $66$ and $66^{65}$ are both even) so $66^{66}\div2$ is even so the last digit must be $8$, not $3$.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.