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?...**

## Comments

### Missing totals

These are the totals that Charlie missed: 17 13 14

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### Missing totals

Did Charlie miss any other totals?

How many different pairs of numbers can you select from a set of five numbers?

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### 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 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.

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### 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?

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### A Pattern/Link!

If you divide the sums of the pairs of numbers added together by the sum of the original numbers, you get the amount of numbers take away 1.

E.G: 17, 20, 6, 18, 12

The sum of the sums of the pairs of numbers: 292

The sum of the original numbers: 73

292 Divided by 73 = 4!

The original numbers amount: 5. My Pattern/Link: Amount of numbers takeaway one (5-1)!

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### A Pattern/Link!

Excellent work! What do you think will happen if you look at sets of three numbers, rather than pairs of numbers?

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