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We have 43 men. They meet each other in 7-men groups. Each of them must see other man (in left 42 men) only once. Describe all combinations how they can meet each other.

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If it was 13 men in groups of 4, difference set {0, 1, 3, 9}.
If it was 21 men in groups of 5, difference set {0, 3, 4, 9, 11}.
If it was 31 men in groups of 6, difference set {0, 4, 10, 23, 24, 26}.
If it was 57 men in groups of 8, difference set {0, 1, 6, 15, 22, 26, 45, 55}.

There isn't a handy difference set of length 7.

The best known answer has 43 meetings before a repeat.

Ed Pegg
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  • could please tell me more details. what you mean in difference set? –  Jun 03 '16 at 08:13
  • also in this list i see that number 1 has met number 21 twice in row 5 and 7. so its not correct i think –  Jun 03 '16 at 10:44
  • The first difference set gives:
    {{0, 1, 3, 9}, {1, 2, 4, 10}, {2, 3, 5, 11}, {3, 4, 6, 12},
    {4, 5, 7, 0}, {5, 6, 8, 1}, {6, 7, 9, 2}, {7, 8, 10, 3}, {8, 9, 11, 4}, {9, 10, 12, 5}, {10, 11, 0, 6}, {11, 12, 1, 7}, {12, 0, 2, 8}} https://en.wikipedia.org/wiki/Difference_set
    Looks like http://www.ccrwest.org/cover.html might have an error for that one.     – Ed Pegg Jun 03 '16 at 19:53
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    The La Jolla covering design repository has been migrated by Dan Gordon to his new site, so it might be a good time to update your link. – hardmath Mar 16 '20 at 00:42