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12 chess players took part in a tournament. Each played against each other exactly once. After the tournament every chess player did 12 lists of names.

  • On the first list, the player only wrote his own name.
  • On the second list, they wrote their own names as well as all man they had won against.
  • They proceeded to write lists: Every next list contains all names from the previous list and the new names that players from the previous list had won against.

It turned out that for all chess players, the 11th and the 12th list contained different amount of each name.

How many games ended draw in the tournament?

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The number of draws must have been

54

In order for the lists to keep growing right up until the end, each list must contain exactly one more person than the previous (they can't grow by more than that, otherwise the last list would have to have more than 12 names).

In particular, the second list must contain exactly two people, meaning that each person in the tournament beat exactly one other person. This means there were only 12 victories total in the tournament. Subtracting this from the total number of matches, 66, gives the number of draws.

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