### 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?

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

### Have You Got It?

Can you explain the strategy for winning this game with any target?

##### Age 11 to 14 Challenge Level:

List any 3 numbers.

It is always possible to find a subset of adjacent numbers that add up to a multiple of 3 (that is either one, two or three numbers that are next to each other). For example:

 5, 7 , 1 5 + 7 = 12 (a multiple of 3) 4,4, 15 15 is a multiple of 3 5,11,2 5 + 11 + 2 = 18 (a multiple of 3)

Can you explain why and prove it?

What happens if you write a list of 4 numbers?
Is it always possible to find a subset of adjacent numbers that add up to a multiple of 4?
Can you explain why and prove it?

What happens if you write a long list of numbers (say n numbers)?
Is it always possible to find a subset of adjacent numbers that add up to a multiple of $n$?
Can you explain why and prove it?