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Let's say there are 10 golfers playing a mini 3 round tournament, 1 round on each of 3 consecutive days. Each day they are split in to 2 groups of 5 golfers. Is it possible that, by the time the tournament is over, each golfer has played with each other golfer at least once?

Follow up question: If it is not possible, what is the largest grouping size for which it is possible?

Edit: By "largest grouping size" I mean, what is the largest N such that N golfers can be split in to 2 groups of N/2 each day and by the end each of the N golfers will have played with each other golfer.

nurdyguy
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  • With 8 golfers, they can first split into four pairs, with tournament {AB, CD}, {AC, BD}, {AD, BC}. – vadim123 Oct 23 '18 at 16:21
  • With 9 golfers, this is impossible, as 9 is odd. Hence it only remains to prove that it's not possible with 10 golfers. – vadim123 Oct 23 '18 at 16:32
  • Simulations give N=4. Even for 6 golfers there are at least two pairs that wouldn't play in 3 days. – Vasily Mitch Oct 23 '18 at 16:51
  • @vadim123 Your example seems to me to be for N=4 not N=8. Am I missing something? – nurdyguy Oct 23 '18 at 17:01
  • The golfers split into 4 pairs: A, B, C, D. A=1+2, B=3+4, C=5+6, D=7+8. – vadim123 Oct 23 '18 at 17:16
  • Gotcha. Sorry, I thought when you said pairs that AB was the pair not that each of A and B represents a pair. Interestingly though, does this not prove that it is always possible for any N divisible by 4? – nurdyguy Oct 23 '18 at 17:26

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