Treebeard's Stumper Answer 9 February 2001

Round Robin

The Santa Ynez Valley Junior High League boys' basketball tournament starts after school today, following a round robin season where each team played every other team. There's an art to winning games of course, and there's also an art to scheduling games. Can you make such a round robin schedule for eight teams, so that each team plays all the others in seven successive rounds? The challenge is to find a method or algorithm that works for any number of teams. Could we schedule a massive round robin at school so that each of the 66 DMS students could interview every other student on 65 consecutive days?

Math problems come in many guises. Years ago, Dunn Middle School Director John Seigel-Boettner wanted to set up a round robin schedule to have each student interview every other on successive days. We struggled to create that schedule by trial and error and finally gave up. It is possible with the right method or algorithm, such as the polygon method shown below. This method also works for an odd number of teams if you replace the middle team with a "bye." It's hard to visualize rotating a 66-side polygon, but it's a snap for the small computer program I wrote.

Here's a round robin tournament schedule for 8 teams. The teams (1 - 8) are listed across the top, and their opponents for each round are listed in the table below, e.g. in round 3, team 1 plays team 4 (and vica-versa). You can find all these matches on the clockwise rotating heptagon shown below!
teams:	1	2	3	4	5	6	7	8
round 1:	8	7	6	5	4	3	2	1
round 2:	6	5	4	3	2	1	8	7
round 3:	4	3	2	1	7	8	5	6
round 4:	2	1	7	6	8	4	3	5
round 5:	7	6	5	8	3	2	1	4
round 6:	5	4	8	2	1	7	6	3
round 7:	3	8	1	7	6	5	4	2

Notes:

Every craftsman has a toolkit, and of course the many different crafts need different tools. Scheduling problems like this require a toolkit of algorithms to get the job done. Algorithms are a cultural legacy as vital as a wrenches and hammers and logic probes! John and I tried to solve a massive scheduling problem at school by trial and error without the right tools, and we gave up because we didn't know the right method.

You can rotate a 7-sided heptagon with a stationary center point to produce a round robin schedule for 7 or 8 teams. For an odd-number of teams, just replace any team with a "bye", so whoever plays that team sits out the round. It's interesting how geometry provides the answer to this completely different problem!

You can rotate the polygon to make a schedule, but it comes to the same thing if you keep the polygon fixed and rotate the teams. That's easier to do on paper or in a computer program, and it really comes to the same thing if you think about it.

Line up the contestants facing each other. If there's an odd number of teams, just add a dummy team to get to an even number and consider it a "bye". Whoever faces the imaginary "bye" team sits out that round. With eight teams, it looks like this, same as above:

     1   2   3   4
    *8*  7   6   5

For each successive round, one team stays fixed (say team *8* in this example), and the others move to the next place clockwise:

     7   1   2   3       6   7   1   2       5   6   7   1     
    *8*  6   5   4      *8*  5   4   3      *8*  4   3   2 

     4   5   6   7       3   4   5   6       2   3   4   5     
    *8*  3   2   1      *8*  2   1   7      *8*  1   7   6 

After these seven steps, everyone is back to where they started, and we're done. With even n or odd n-1 teams, we're done after n-1 rotations.

The teams can rotate clockwise (as above) or counterclockwise, it doesn't matter. And any one of them can stay fixed, that doesn't matter either. And the numbers can be in any order besides the clockwise 1 to 8 shown above. If there are an odd number of teams, then just add a "dummy" team to get to an even number, and whoever faces the dummy sits out that round. It's convenient if the dummy is the *fixed-position* team, but any place will do.

An interesting stumper remains. How many uniquely different round robin schedules are possible for any n teams? It might be easiest to think of this as a geometry problem. How many unique mappings are there between pairs of vertices on an n-sided polygon such that no pairings are left out or repeated after n-1 rotations? Any n-sided polygon has n edges and n(n-3)/2 diagonals.

John, here's your schedule for a round robin of (now) 66 kids interviewing each other over 65 successive days.
Sorry it's a decade too late!

Round Robin Interviews with 66 people:																																																																		
Scheduled by ROBIN.BAS, 17 Feb 2001
                                                                                                                                    
Team: 1   2   3   4   5   6   7   8   9   10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  66
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Round\
  1:  66  65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1
  2:  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   66  65
  3:  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  66  63  64
  4:  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  66  62  61  63
  5:  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  66  61  60  59  62
  6:  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  66  60  59  58  57  61
  7:  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  66  59  58  57  56  55  60
  8:  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  66  58  57  56  55  54  53  59
  9:  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  66  57  56  55  54  53  52  51  58
 10:  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  66  56  55  54  53  52  51  50  49  57
 11:  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  66  55  54  53  52  51  50  49  48  47  56
 12:  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  66  54  53  52  51  50  49  48  47  46  45  55
 13:  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  66  53  52  51  50  49  48  47  46  45  44  43  54
 14:  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  66  52  51  50  49  48  47  46  45  44  43  42  41  53
 15:  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  66  51  50  49  48  47  46  45  44  43  42  41  40  39  52
 16:  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  66  50  49  48  47  46  45  44  43  42  41  40  39  38  37  51
 17:  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  66  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  50
 18:  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  66  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  49
 19:  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  66  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  48
 20:  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  66  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  47
 21:  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  66  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  46
 22:  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  66  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  45
 23:  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  66  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  44
 24:  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  66  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  43
 25:  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  66  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  42
 26:  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  66  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  41
 27:  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  66  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  40
 28:  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  66  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  39
 29:  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  66  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  38
 30:  8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  66  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   37
 31:  6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  66  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   36
 32:  4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  66  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   35
 33:  2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  66  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   34
 34:  65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  66  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   33
 35:  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  66  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  32
 36:  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  66  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  31
 37:  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  66  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  30
 38:  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  66  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  29
 39:  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  66  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  28
 40:  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  66  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  27
 41:  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  66  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  26
 42:  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  66  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  25
 43:  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  66  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  24
 44:  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  66  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  23
 45:  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  66  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  22
 46:  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  66  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  21
 47:  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  66  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  20
 48:  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  66  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  19
 49:  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  66  17  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  18
 50:  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  66  16  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  17
 51:  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  66  15  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  16
 52:  29  28  27  26  25  24  23  22  21  20  19  18  17  16  66  14  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  15
 53:  27  26  25  24  23  22  21  20  19  18  17  16  15  66  13  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  14
 54:  25  24  23  22  21  20  19  18  17  16  15  14  66  12  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  13
 55:  23  22  21  20  19  18  17  16  15  14  13  66  11  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  12
 56:  21  20  19  18  17  16  15  14  13  12  66  10  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  11
 57:  19  18  17  16  15  14  13  12  11  66  9   8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  10
 58:  17  16  15  14  13  12  11  10  66  8   7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  9
 59:  15  14  13  12  11  10  9   66  7   6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  8
 60:  13  12  11  10  9   8   66  6   5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  7
 61:  11  10  9   8   7   66  5   4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  6
 62:  9   8   7   6   66  4   3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  5
 63:  7   6   5   66  3   2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   4
 64:  5   4   66  2   1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   3
 65:  3   66  1   65  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6   5   4   2

The right algorithm can solve this stumper, but another puzzle remains. Why do we call this a round robin schedule? It has nothing to do with fat red-breasted birds!

The Usenet group uk.culture.language.english explains it like this:

The "Round-robin" almost certainly comes from the French "ruban", a round ribbon, and tradition has it that its English usage in began in the British navy (although citations occur as early as the 16th century in other contexts). This was a way for British tars to sign a petition of grievances in such a way that they could not be identified as the instigator. A "round-robin" was signed in circular fashion, each signature beginning at the center of the circle and extending out to the radius of the circle. It is used in modern English to describe a competition where each of the entrants competes at least once with every other entrant.

A round circle of ribbon would be a fine way to implement the polygon-method among a group of real people seated in opposing rows of real chairs, with an empty chair for the possible "bye". There would be evenly spaced marks on the ribbon, and all but one of the contestants would pass it around from mark by mark. Is it a coincidence that "round" also means complete or full, as in "to round off a number" or "a round dozen"? The use of the phrase "round robin" in sports schedules only dates back to the 1800s, so the etymology should be known, but it's not. I wonder who first came up with this non-obvious algorithm that's now part of our toolkit?

Here are some some Web links for your own research.

• I wrote a small DOS BASIC program that implements the polygon method to figure round robin schedules for any number of teams. It's a quickie, but it works. You can download my ROBIN.BAS program with source and executable from Treebeard's Basic Vault.

• The "polygon method" for figuring round robin tournaments is explained at The Math Forum: Ask Dr. Math, along with this variation and another one. There's another scheduling problem at Macalester College Problem of the Week (POTW). Adam Florence has a different algorithm to solve these problems. There's info on the geometry on the Polygons page.

• There's practical advice on creating round robin schedules at World Team Tennis (WTM) site, and ready-made schedules at UC Berkeley Quiz-Bowl. Real team scheduling is often more difficult because of limited courts and the need to balance who plays who, when. Home or Away is a shareware sports scheduling program from Australia that can deal with the complexities.

• The exact etymology of the phrase "round robin" seems to be unknown, but it's a Frequently Asked Question on word origin Web sites. It's discussed at Wilton's Word & Phrase Origins, Take Our Word For It , uk.culture.language.english, Philoman's Cool Words, Merriam-Webster OnLine, and The Word Detective by Evan Morris. The VERBATIM (Summer 99) ezine adds an interesting extension to the meaning for teens, from the TV show Buffy the Vampire Slayer: "To do a round robin which ... is where everybody calls everybody else's mom and tells them they're staying at everybody's house."

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