### Geoboards

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

### Tiles on a Patio

How many ways can you find of tiling the square patio, using square tiles of different sizes?

### 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?

# Weekly Problem 43 - 2009

##### Stage: 2 and 3 Short Challenge Level:

$ABEF$ is a square, and its area is $4\times$ area$(BDA) = 4\times\frac{1}{2}(BD\times DA) = 2BD^2 = 24$cm$^2$, so $BD=\sqrt{12}=2\sqrt{3}$cm.

The angle $ABD$ is $45^{\circ}$ so the angle $CBD$ is $45^{\circ}-15^{\circ}=30^{\circ}$.

Therefore $\tan{30^{\circ}}=\frac{1}{\sqrt{3}}=\frac{CD}{BD}=\frac{CD}{2\sqrt{3}}$ so $CD=2$cm.

This problem is taken from the UKMT Mathematical Challenges.

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