The Mathematician's Arm Wrestle Tournament
The Mathematician's Arm Wrestle Tournament is an arm wrestling tournament that takes place every year between the world's strongest mathematicians. It is a round-robin tournament where each competitor plays every other exactly once and each game can end in only a win or a loss for a given player.
In 2015, seven mighty mathematicians entered the tournament. The table below shows a list of possible scoresheets containing the number of wins for each competitor.
| Alice | Bob | Carly | Dave | Evelyn | Frank | Greg | |
| Scoresheet 1 | 5 | 2 | 1 | 0 | 2 | 5 | 6 |
| Scoresheet 2 | 5 | 4 | 3 | 3 | 3 | 2 | 2 |
| Scoresheet 3 | 3 | 3 | 3 | 3 | 3 | 0 | 6 |
| Scoresheet 4 | 5 | 4 | 1 | 1 | 4 | 0 | 6 |
Which of the score sheets reflects possible scores for that year?
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Score Sheet 3 is the only one that works, and it is not hard to come up with a tournament with these scores.
In Score Sheet 1, player G won 6 games, which means he won all of his games. Players A and F won 5 games, so the only games they lost were their games against player G. But players A and F also played against each other, so one of them must have lost another game. In Score Sheet 2, the scores add up to 22, but the scores must always add up to ( 2 7 ) = 2 1 .
In Score Sheet 4, players C, D, and F have a combined total of 2. But these three players played against each other (in 3 games), so their combined total has to be at least 3.
For a tournament, the numbers on a score sheet form what is called a "score sequence" (not surprisingly). It turns out there is an easy way of telling whether a sequence of numbers can be a score sequence for a tournament. See https://en.wikipedia.org/wiki/Tournament (graph theory)#Score sequences and score sets for more details.