You may wish to take a look at Pairwise Adding before trying this problem
Alison reckons that if Charlie gives her the sums of all the pairs of numbers from any set of five numbers, she'll be able to work out the original five numbers.
Charlie gives her this set of pair sums: $0$, $2$, $4$, $4$, $6$, $8$, $9$, $11$, $13$, $15$.
Can you work out Charlie's original five numbers?
Can you work out a strategy to work out the original five numbers for any set of pair sums that you are given?
Does it help to add together all the pair sums?
Given ten randomly generated numbers, will there always be a set of five numbers whose pair sums are that set of ten?
Can two different sets of five numbers give the same set of pair sums?
Four numbers are added together in pairs to produce the following answers: $5$, $9$, $10$, $12$, $13$, $17$.
What are the four numbers?
Is there more than one possible solution?
Six numbers are added together in pairs to produce the following answers: $7$, $10$, $13$, $13$, $15$, $16$, $18$, $19$, $21$, $21$, $24$, $24$, $27$, $30$, $32$.
What are the six numbers?
Can you devise a general strategy to work out a set of six numbers when you are given their pair sums?