Discover Boggler, June 2001


	THE DISCOVER BOGGLER Jun 2001: And the Winner Is...

I was pretty appalled by the confusion around the 2000 US presidential elections. But out of confusion came a good subject for a puzzle. Unfortunately it takes about five months for one of my puzzles to appear in Discover, so by the time the puzzle came out the election was a thing of the past. My editors recommended I not focus too much on the presidential elections, and also include other sorts of contests.

Electoral Confusion. It is possible, though unusual, for a US presidential candidate to win the popular vote but lose the electoral college. This puzzle attempts to explain this weird mathematical anomoly, by exploring just how wide the gap can be.

Flavor Face-Off. While researching this puzzle I found that Discover itself had run an article "May the Best Man Lose" about election methods in the November 2000 issue.

Tennis Mismatch. I researched this puzzle by searching for computer programs that schedule tennis tournaments. I found many programs, including one that claimed to be the only program that correctly schedules a mixed double round robin in which every player plays with every other player, against every other player. Fascinating stuff, but too hard for a magazine puzzle.
For space reasons my editors omitted a fourth Tennis Mismatch puzzle. It's quite beautiful mathematically, but has little to do with the previous three puzzles. Here it is:

4. Judge Judy wants a round robin tournament, so she can rank all the players 1 through 8. Each day every player plays one game. At the end of seven days every player has played every other player. There are four courts. Can you make a schedule that meets these requirements? Can you come up with a systematic way to schedule round robin tournaments with any number of players?

And the answer is

4. Here’s a schedule for a seven day round robin tournament.

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

To make this schedule, draw seven dots in a heptagon, and add one more dot in the middle. Connect the dots with lines as shown below.

This represents the first day’s matches. Each day, rotate the lines clockwise to the next number. The line to 1 pivots around the center. The same method works for any even number of players. For odd numbers, omit the center player, and let the unmatched player rest that day. This method appears in Ringel and Hartsfields’ Pearls of Graph Theory.