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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 Short
Challenge Level

Answer: 8

$6^1 = 6$
$6^2 = 36$
$6^3 = 216$
$6^4 = ...6$ 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.