Pairwise Adding


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?...



These are the totals that Charlie missed: 17 13 14

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

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.

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?

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)!

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