An easy example we have used elsewhere is 16 golfers playing 5 rounds of golf. Think of yourself as one of the golfers. In 5 rounds of golf, you will have three others in your foursome over 5 rounds, for a total of 15 companions. Besides you, there are 15 other golfers. Aha – you should play with every other golfer exactly once. And if every one of the 16 golfers has this profile, then we have a perfect pairing. The number of unique ways to place 16 golfers into four groups over 5 rounds is 10 times a trillion times a trillion, so this truly is looking for a needle in a haystack.

Sometimes a perfect solution is not mathematically possible. One of life’s cruel jokes is that the most common trip size is twelve golfers and 6 rounds, and it is not possible to generate a perfect pairing for 12 golfers over 6 rounds. Again, let’s consider one of these 12 golfers. In six rounds, this golfer will have six times three, or 18 companions. There are 11 other golfers. If there are 11 golfers and 18 “slots” to fill, in a perfect world a golfer would be with seven of the 11 golfers twice, and four golfers once (seven times 2 is 14 and 4 times 1 is 1, for a total of 18). If every golfer had this profile, we would have a “perfect pairing”. Alas, this is not possible. There are two reasonable choices. If we want everyone to play with everyone else at least once, then there will be exactly 3 cases where two golfers are paired three times. If you do not want any cases of players being paired together three times, this is possible, but there will be three cases where two players are not paired together at all. Choose you poison. Our scheduler provides a “switch” to choose your preference.

Fortunately, in most cases we can find a perfect pairing. Eight players playing 6 rounds is such an example. In six rounds, a golfer will be with 18 companions, as above. Since there are only 7 other golfers, ideally every golfer should be paired with four golfers 3 times and three golfers 2 times. We can find such a perfect pairing in less than a second.

This all gets really interesting and challenging when we start to place demands on the schedule – golfers missing from rounds, golfers who should be paired together in certain rounds, golfers who should not be paired together in certain rounds, etc. Let’s consider the same tournament style that is discussed in the Sneak Peek section of the site. We have 12 golfers playing 6 rounds. In rounds 2 and 3 we want a team format for a mini Ryder Cup, so everyone has the same partner in rounds 2 and 3. We team an A with a D and a B with a C. In rounds 4 and 5, we are playing best net two balls of foursome, so we want each foursome to have one A, one B, one C and one D. The scheduler produces a schedule that is not quite as good as 12 golfers and 6 rounds with no constraints. As we stated above, in the simple case of 12 golfers and 6 rounds, we would have 3 cases of two players being paired 3 times. In this case, with the team play in rounds 2 and 3, and ABCD format in rounds 4 and 5, we have 6 cases of two players being paired three times instead of just 3 cases. The other 60 player 1/ player 2 pairing counts are all 1 or 2. Pretty darned good. If someone can do better, we’d love to hear about it.