You Only Have Their Sums

Algebra Level 4

There are \(n \geq 3\) people sitting in a circle, and each person is thinking of a number. Every consecutive pair announces the sum of their numbers (but not the individual numbers). For example, if there are three people with numbers \( a, b \) and \( c \) in that order, the three sums announced are \( a + b, b + c, \) and \( c + a .\)

For what values of \(n\) can an outsider deduce all of the individual numbers?

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