Say you’ve got about 25 members of your clan descending on your city for a family reunion, and you want to have a tournament be part of the entertainment. There will be a few new in-laws this year, and you want them to have a chance to meet everyone. Individual games don’t take very long. So make it a round robin!

A daunting organizational task you think. But it’s actually quite doable. I’ve got your format.

The most fun I ever had playing a tournament of any kind was at a fencing tournament held in the dead of winter in Silverton, Colorado in early 1975.

Silverton is in the rugged mountains of southwestern Colorado. It’s a beautiful area that hosted a fair number of tourists in the summer. In the dead of winter the town mostly shut down. But a couple of the hardy locals who lived there year round were avid fencers, and they organized a tournament. They’d invited fencers from the surrounding area for a weekend of competition, and persuaded one of the hotels to open for a few days so they’d have a place to stay. One of them had a big empty house where the impecunious folks like me and other members of the St. John’s College fencing club could unroll a sleeping bag.

They laid out pistes in various parts of town. One, I remember, was set up parallel to the bar in a tavern catering to the rough miners who were most of Silverton’s winter inhabitants. There was so little room in the bar that the director had to climb up into the rafters and officiate looking down at the fencers. The fencers in their whites pranced back and forth as a few grizzled miners looked on with tolerant amusement.

But the really striking thing about the tournament was that it was run as a complete round robin. I don’t remember exactly how many fencers there were, but is was something like 25. There were several rounds, and for each round they dispatched a small group to one of the venues with instructions to run a round robin among that group, and then report back.

I marveled at the time that this was possible – how can you chop a big round robin up into a lot of little ones? When I returned to Santa Fe, I figured it out. Or rather, I figured out how it probably should have been done. As I recall, in the latter sessions in Silverton, we started being grouped with some folks we’d already fenced, and there were a few fencers at the end who had to seek out an opponent or two that they’d never been grouped with.

The tables at the right show a way to break up a 25-team complete round robin into six five-person round robins with no overlaps and no gaps. The tables are easy to produce once you see the pattern. You start with your entries in numerical order, as in the top table. Then, for each subsequent table, you read the long diagonal into the first row, and then read the other entries down in numerical order, wrapping around when you get to the end of a quintile. It’s easier done than said – just look carefully at the table and you’ll spot the pattern, and be able to adapt it, if need be, to 16 teams, or 36.

I call this the Silverton system in honor of that tournament. If it’s known to others, and has some other name, I’d be glad to learn it.

Unfortunately, you can’t adapt the pattern to 16 or 36. In the 4×4 case, basing the diagonals off of “1”, “11” occurs as the third term in both the first (1,6,11,16) and third (1,8,11,14) diagonals. Likewise, “22” also occurs as the fourth term in the first, third, and fifth diagonals of the 6×6 pattern (1,8, 15, 22, 29, 36; 1,10,13,22,25,34; and 1,12,17,22,27,32) respectively.

It does work for 3×3, though, and I would imagine that 7×7 can be utilized this way as well. I believe that the underlying mathematical concepts are the same as the ones for projective planes, with the points at each plane’s respective “line at infinity” representing a time coordinate, such that the other “lines” (sets of rounds) that intersect each of those points happen simultaneously (because those lines are otherwise parallel). Unfortunately, as referenced in Ian Stewart’s “Another Fine Math You’ve Gotten Me Into” (1992), the 6×6 case was proven unworkable by brute force around 1900 and by formula in 1949.

I was, however, able to brute-force the 4×4 table in a “checkerboard” style: that is, terms that would be red/white when the cells are laid on a checkerboard would follow one pattern while terms that would be black would follow the mirror image of that pattern. This is not inconsistent with Stewart, as he did claim that Projective Planes exist for all prime powers, but did not go into detail. This includes the cases for 8×8 and 9×9, but at that point, the gathering in question has likely gotten too large for individual games anyway.

Besides, while it may not be the best at having everyone meet up, I would rather have a Swiss-style Bridge/Euchre tourney for those larger gatherings…

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