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Question from Brian:

We have a hockey roster of 19 kids. We only want to play 15 kids for each game. 4 players will not play each game. There are 16 games total. We want to rotate who doesn't play evenly. Can this be done? How many games will each player sit during the season?


If 4 players miss each game, then 64-player-games are missed over the 16-game season. That means the best possible outcome is that each player will miss at least three games, and 7 players will miss four games.

Here is a way to do it. Write the names of the players in a list 1 through 19. Then list them again (maybe in the same order) so that the player in position 20 is not one of the ones in position 17, 18, 19. Do it again, and make sure that the players in positions 39 and 40 are not also in position 37 or 38. Now choose 7 players who are different then the one in position 57 and add their names to the list. These are the players who will sit out 4 games.

Now, running down your list, the first 4 players miss game 1; the next 4 miss game 2; and so on.

A good way to make the list might be to put the 19 names in a hat and draw them without replacement, then put them back and repeat the process. If a name that’s drawn violates one of the constraints (e.g. the player in position 20 is one of the ones in position 17, 18, 19), then just put it back in the hat and draw again.



No, it cannot be done. There are a total of 64 cases of players sitting out a game, and 19 does not divide that evenly. At best, 12 kids will sit out 3 games and 7 will sit out 4.


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