### 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?

### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

### Sweets in a Box

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

# Route Product

##### Age 7 to 11Challenge Level

There are lots of different routes from $A$ to $B$ in this diagram:

The idea is to work out the product of the numbers on these different routes from $A$ to $B$. Let's say that in a route you are not allowed to visit a point more than once.

For example, we could have $3\times0.5$ but we couldn't have $3\times2\times5\times4\times1\times 0.1$ because that route passes through $A$ twice.

Which route or routes give the largest product?

Which route or routes give the smallest product?

Do you have any quick ways of working out the products each time?

[This problem is adapted from a SMILE Centre card.]