Charlie chose five numbers: $2$, $3$, $4$, $7$, $10$.

He added together pairs of numbers from his set, and got the following totals: $5$, $7$, $9$, $10$, $11$, $12$.

Can you find the totals that Charlie has missed?

Choose your own set of five numbers, and work out all the possible sums of pairs.

How do you know you've found them all?

How many pair sums would there be if you started with six numbers? Or seven numbers? Or ...?

Can you find a set of five different numbers, where the sums of some pairs give the same total?

Can you find a link between the total of any set of five numbers, and the total of their pair sums?
What if you started with a set of six numbers? Or seven? Or ...?

What happens when you choose to add sets of three numbers instead of pairs?...

### Missing totals

These are the totals that Charlie missed: 17 13 14

### Missing totals

Did Charlie miss any other totals?
How many different pairs of numbers can you select from a set of five numbers?

### For a set of 1 number there

For the number of pair sums for a set of numbers, we have:

For a set of 1 number there are 0 pairs
For a set of 2 numbers there is 1 pair
For a set of 3 numbers there are 3 pairs

For a set of 4 numbers there are 6 pairs
For a set of 5 numbers there are 10 pairs
For a set of 6 numbers there are 15 pairs
For a set of 7 numbers there are 21 pairs
For a set of 8 numbers there are 28 pairs

If we set this out as a sequence it would be
0, 1, 3, 6, 10, 15, 21, 28 .......................

The nth term of this sequence is $0.5 (n^2 - n)$ where n is the number of numbers.

### For a set of 1 number there

Very good work, Sam!

Can you find any connection between the total of a set of numbers and the total of the pair sums for that set of numbers?