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So, given any set of three integers, prove there is a pair whose sum us even, and then prove or disprove that there is a pair whose sum is odd.

To prove that there is a pair whose sum is even, couldn't I say that since there are 3 integers that are either even or odd, there must be 2 that are even, or 2 that are odd, in which the sum of the even pair or odd pair is even?

For the second part, I know that there can be a few possibilities for an odd sum, but that is dependent upon the set of integers, so I'm not sure how to exactly prove that.

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  • $\begingroup$ The question seems a little ill-worded- is it supposed to read along the lines of: 'there must be a pair whose sum is even/odd'? $\endgroup$ – Sherlock Holmes Oct 1 '14 at 4:50
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According to pigeonhole principle, since there are $3$ integers (pigeons) and each of which can be even or odd ($2$ holes), either there are $\lceil \dfrac{3}{2}\rceil = 2$ odd or $2$ even integers, both of which give an even sum.

However, there is a possibility that all are even, therefore a pair of integers with odd sum can't be guaranteed.

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