My eight cousins are all a different prime number of years old, and their average age is a whole number. Recently they all came to visit me and, as they left one by one, I noticed that at all times the average age of the cousins remaining was also a whole number. Moreover, I noticed that the sum of their ages was the least it could have been for this to happen.

How old are my eight cousins?


Their ages are


which is a combined total age of


This is one possible order in which they left (although not unique)

13, 7, 19, 5, 3, 11, 17, 29

How did I solve this

None of their ages can be 2 since the overall sum must be even. The smallest sum of primes is then 3+5+7+11+13+17+19+23 = 98 but this is not divisible by 8. The next smallest is the solution given (which is divisible by 8) and it is not too difficult to work out an order in which they must leave to satisfy the constraints.

  • $\begingroup$ lol, expecting something original when I first read the question... next try and the answer satisfies :) $\endgroup$ – Oray Jan 26 at 19:46
  • $\begingroup$ @Oray Agreed, at first I thought there might be a bit more searching to do. $\endgroup$ – hexomino Jan 26 at 19:47
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    $\begingroup$ @hexomino Had my cousins been ten instead of eight? $\endgroup$ – Bernardo Recamán Santos Jan 26 at 19:58
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    $\begingroup$ @BernardoRecamánSantos The case of ten is very interesting a does require a little more legwork. I think the answer is rot13(bar-uhaqerq naq avargl. Bar-uhaqerq naq friragl vf gur fznyyrfg cbffvovyvgl nf n fhz bs gra cevzrf ohg frrzf gb snvy fhofrdhragyl naq bar-uhaqerq naq rvtugl vf qvivfvoyr ol obgu avar naq gra juvpu ehyrf vg bhg. Bar-uhaqerq naq avargl vf qbnoyr gubhtu.) $\endgroup$ – hexomino Jan 27 at 10:34

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