64 Teams, rounds with groups of 8 selected players

[Bartholomew]

Greetings everyone,

I'm presented with quite a unique challenge I've been trying to wrap my head around for a while now. Clearly I have failed and I was wondering if you'd be so kind as to give me some input. I'm currently organizing a tournament for teams of graphic artists. In total there are 64 teams. The concept is to have these 64 teams battle it off in groups of 8. However the condition is teams should never have to face the same opponent teams twice. In its most basic form it would create 8 divisions per round, consisting of 8 teams.

An example of this for Round 1 of the tournament would be
Division #1: Teams 1 to 8
Division #2: Teams 9 to 16
.. And so on for 8 divisions.

The next round the divisions would be shuffled with new, unique team combinations.

I've gone as far as making a 64 cell quadrant and using a striping method as well as trying formulas in excel. I've gotten quite far and in most cases I know what the cause of duplicates is.. I just have no idea how to fix it.

So on to the meat and potatoes, I was wondering if you could assist me in generating as many rounds as possible that meet the condition (if I understood correctly from the theory, there should be 7 possible rounds?), preferably in Excel format. Having my brain trying to process this for the last few days has made me quite eager to get some more understanding of it, so if you could provide the supporting mathematics or programming that'd be a great bonus. That being said, any help would be much appreciated!

[Ian Wakeling]

It's possible to have as many as 9 rounds at which point every team will have been
placed in a division with all 63 other teams exactly once.

In mathematical terms you need to construct an order 8 affine plane, see here for diagrams of the order 2 and 3 planes.  For the construction itself you will need to use a Galois Field.  Each line connects all the teams in one division and each set of 'parallel' lines forms one round.

It should be possible to copy/paste the schedule below on to an Excel worksheet and then use a fixed width 'Text to Columns' on it to get 10 separate columns of numbers.

Hope that helps.

Round Div          Teams
   1   1   ( 1  2 47 11 12 38 48 37)
   1   2   ( 3  4 45  9 10 40 46 39)
   1   3   ( 5  6 43 15 16 34 44 33)
   1   4   ( 7  8 41 13 14 36 42 35)
   1   5   (17 18 63 27 28 54 64 53)
   1   6   (19 20 61 25 26 56 62 55)
   1   7   (21 22 59 31 32 50 60 49)
   1   8   (23 24 57 29 30 52 58 51)

   2   1   ( 1 64 42  3 62 21 23 44)
   2   2   ( 2 63 41  4 61 22 24 43)
   2   3   ( 5 60 46  7 58 17 19 48)
   2   4   ( 6 59 45  8 57 18 20 47)
   2   5   ( 9 56 34 11 54 29 31 36)
   2   6   (10 55 33 12 53 30 32 35)
   2   7   (13 52 38 15 50 25 27 40)
   2   8   (14 51 37 16 49 26 28 39)

   3   1   ( 1 45 56 32 52  5 28 41)
   3   2   ( 2 46 55 31 51  6 27 42)
   3   3   ( 3 47 54 30 50  7 26 43)
   3   4   ( 4 48 53 29 49  8 25 44)
   3   5   ( 9 37 64 24 60 13 20 33)
   3   6   (10 38 63 23 59 14 19 34)
   3   7   (11 39 62 22 58 15 18 35)
   3   8   (12 40 61 21 57 16 17 36)

   4   1   ( 1 55 50 58 16 63  8  9)
   4   2   ( 2 56 49 57 15 64  7 10)
   4   3   ( 3 53 52 60 14 61  6 11)
   4   4   ( 4 54 51 59 13 62  5 12)
   4   5   (17 39 34 42 32 47 24 25)
   4   6   (18 40 33 41 31 48 23 26)
   4   7   (19 37 36 44 30 45 22 27)
   4   8   (20 38 35 43 29 46 21 28)

   5   1   ( 1 15 17 14  4 20 31 30)
   5   2   ( 2 16 18 13  3 19 32 29)
   5   3   ( 5 11 21 10  8 24 27 26)
   5   4   ( 6 12 22  9  7 23 28 25)
   5   5   (33 47 49 46 36 52 63 62)
   5   6   (34 48 50 45 35 51 64 61)
   5   7   (37 43 53 42 40 56 59 58)
   5   8   (38 44 54 41 39 55 60 57)

   6   1   ( 1 61 59 29 33 27  7 39)
   6   2   ( 2 62 60 30 34 28  8 40)
   6   3   ( 3 63 57 31 35 25  5 37)
   6   4   ( 4 64 58 32 36 26  6 38)
   6   5   ( 9 53 51 21 41 19 15 47)
   6   6   (10 54 52 22 42 20 16 48)
   6   7   (11 55 49 23 43 17 13 45)
   6   8   (12 56 50 24 44 18 14 46)

   7   1   ( 1 13 46 26 22 57 34 53)
   7   2   ( 2 14 45 25 21 58 33 54)
   7   3   ( 3 15 48 28 24 59 36 55)
   7   4   ( 4 16 47 27 23 60 35 56)
   7   5   ( 5  9 42 30 18 61 38 49)
   7   6   ( 6 10 41 29 17 62 37 50)
   7   7   ( 7 11 44 32 20 63 40 51)
   7   8   ( 8 12 43 31 19 64 39 52)

   8   1   ( 1 36 10 25 60 51 43 18)
   8   2   ( 2 35  9 26 59 52 44 17)
   8   3   ( 3 34 12 27 58 49 41 20)
   8   4   ( 4 33 11 28 57 50 42 19)
   8   5   ( 5 40 14 29 64 55 47 22)
   8   6   ( 6 39 13 30 63 56 48 21)
   8   7   ( 7 38 16 31 62 53 45 24)
   8   8   ( 8 37 15 32 61 54 46 23)

   9   1   ( 1 54 35 40 19 49 24  6)
   9   2   ( 2 53 36 39 20 50 23  5)
   9   3   ( 3 56 33 38 17 51 22  8)
   9   4   ( 4 55 34 37 18 52 21  7)
   9   5   ( 9 62 43 48 27 57 32 14)
   9   6   (10 61 44 47 28 58 31 13)
   9   7   (11 64 41 46 25 59 30 16)
   9   8   (12 63 42 45 26 60 29 15)

[Bartholomew]

You're an absolute lifesaver, much appreciated.