I've tried asking this question of various mathematicians, but never gotten an answer. Perhaps because it is not well posed. I'll present the problem, but it may be that certain simplifying assumptions must be made in order to get a handle on it.

At the Manhattan Scrabble Club four games are played by each player during an evening tournament. Each game is played by two players. (If there are an odd number of players present on a given night, then the moderator plays too, otherwise he doesn't play.) There is a slight advantage to the player who plays first, so it is desirable that no player should play first more times than his opponents. For this reason before a game between two players begins, their records for previous games on that night are compared. If they have each played first in the same number of previous games, then each of them selects a tile from the bag without looking. The player who selects the tile earliest in alphabetical order gets to go first in the new game. (If they choose the same letter, then each of them returns his tile to the bag and they choose again until someone prevails.) If one player has played first in more previous games than the other, then his opponent gets to play first in the new game. In actual play there is a moderator who tries to match player according to their records for that night . For example: if, before the fourth game, there are two players each of whom has won all three of his previous games then the moderator will try to match them against each other. Perhaps it would be best to leave out the effect of the moderator initially when trying to analyse the setup.

Despite these measures, it seems clear that it is possible that one or more players will play first in each of his, or their games.

First question: What is the smallest number of players who must be present at the tournament such that it is possible that at least one player plays first in all four of his games?

Second question: Given that the number of players in the tournament is as in the answer to the first question, can we assign a probability to the occurence that on a certain night one or more of the players has played first in all four games? If so, what is that probability?

  • $\begingroup$ @BenjaminDickman I removed the impossible condition that there are two players who have played each other, yet they both have perfect records. Thanks for pointing this out. $\endgroup$ – David S. Newman Dec 16 '13 at 8:24
  • $\begingroup$ Does everyone play their first game at the same time? $\endgroup$ – Calvin Lin Dec 16 '13 at 8:47
  • $\begingroup$ @benjaminDickman It seems that you are right and my question has indeed been migrated, because it is not research level. My favorite teacher, David M. Bloom, once wrote me that he did not consider himself good enough to be a research mathematician. He was a Putnam Fellow. $\endgroup$ – David S. Newman Dec 16 '13 at 8:54
  • $\begingroup$ @DavidS.Newman Problem solving and research mathematics are not one and the same; unsurprisingly, then, they have different websites. Hopefully you get a satisfying answer here! (And for the record: I'm a major Scrabble fan.) $\endgroup$ – Benjamin Dickman Dec 16 '13 at 9:13

Note that you ask for possible, as opposed to guaranteed.

In order for someone to go first in all four games, the two people who play at the final round must have gone first in the first 3 games. .

This in turn implies that there are at least 4 people who have gone first in the first 2 games, and hence at least 4 people who have gone first in the first game and hence at least 16 people.

To show that 16 is enough, we just need to find a construction. This is very easy, by simply letting the person who goes first win the game, we see that the conditions are statisfies.

Hence the answer is 16.

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