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Most Difficult Puzzle

Blueland Soccer Tournament Puzzle

The country of Blueland has four soccer teams that compete for the national championship. The teams’ names correspond to their professions—Miners, Bankers, Police, and Army. (Blueland is a small nation and does not have professional players—all of them have other jobs.) The tournament has two stages. Stage 1 is a round robin in which each team plays against all of the others. Stage 2 is the final match between the two best teams from Stage 1, and it decides the championship. In the event two or more teams have the same number of points at the end of the round-robin stage, a random draw breaks the tie. In the middle 1990s, the country followed the rest of the world and changed the scoring system so that a win is worth three points (instead of two points under the old system). A tie and a loss bring one and zero points, respectively, before and after the change. All four teams are equally strong, and the probability of a tie in any given game is p. The Miners and the Bankers, however, are less than honest and never play competitive games between each other. They always agree on the result before the game begins. Given their records, the probability of a tie in the game between the Miners and the Bankers is q. Answer the following questions:

  1. If there’s a tie between the Miners and the Bankers, what probability p maximizes chances for the Police to make the final game?
  2. If the Miners beat the Bankers, what probability p minimizes chances for the Army to make the final game?
  3. Assuming uniform distribution of p on the interval [0, 1], what probability q triggers the expectation of the Police winning the championship to be equal to 1⁄2?
  4. How do the answers to the previous three questions change if the old scoring system is restored?