### Pair Sums

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

### Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Big Powers

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

# Are We Nearly There?

##### Stage: 3 and 4

Take some time to look at the route followed by the arrows in this diagram.
Can you now look away and list the points visited?

If the pattern of arrows continues for ever, which point will be the $100^{th}$ to be visited?

How many steps will it take to reach (60,40)?
Where will the next step take you to?

Can you design an alternative route that visits all the points on a grid?
Can you still work out how many steps it will take you to reach (60,40)?