### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

### Tri-split

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

# Guillotine

##### Stage: 3 and 4 Short Challenge Level:

$\frac{5}{12}$

Let $AB$ and $AD$ be of length $b$ and $h$ respectively. Then the area of $ABCD = bh$ and the area of $$ABQP = \frac{1}{2}b\left(\frac{h}{2} + \frac{h}{3}\right)= \frac{5bh}{12}$$

Thus the required ratio is $\frac{5}{12}$.

This problem is taken from the UKMT Mathematical Challenges.

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