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# Combinatorics #4

Six distinct positive integers $$a,b,c,d,e,f$$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $$10$$ prime numbers while Jill claims that she has $$9$$ prime numbers among the sums. Who has the correct claim?

Note by Victor Loh
2 years, 9 months ago

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Jack has the wrong claim.

Since the numbers are distinct positive integers, the sum of a pair$$\ge 1+2=3$$, hence the primes must be odd. On the other hand, we can't obtain $$10$$ odd sums because if $$n$$ numbers are odd($$0\le n\le 6$$), then the number of odd sum is $$n*(6-n)$$(we choose an odd then an even), which doesn't equal to $$10$$ for all $$n$$, but $$9$$ primes is obtainable when $$n=3$$. · 2 years, 9 months ago

Can you find a possible set of six positive integers that satisfies the conditions then? · 2 years, 9 months ago

yes it suffices to find 3 sets of three primes that are spaced the same way, so take $$1,3,7,4,16,40$$. · 2 years, 9 months ago