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A Series of Headaches

 Category: Logic Submitted By: eighsse Fun: (2.3) Difficulty: (2.85)

The Detroit Tigers played the Cincinnati Reds in the World Series (Major League Baseball). In this particular series (which went the full seven games), there was only one shutout, and no team scored more than 12 runs in any game. Following are some facts that apply to BOTH teams, individually, pertaining to the number of runs scored in each of the 7 games (no information is given about the order in which these games occured). For each team, there were:

[1] 4 games with a prime number of runs.
[2] A multiple of 3 games with an even number of runs.
[3] No two games, the product of whose runs was a prime number.
[4] 2 games with a perfect square of runs.
[5] No game with exactly 5 runs.

The Tigers scored 30 runs in the entire series, and [6] scored 7 or 8 runs in at least one game. The Reds scored 36 runs, and [7] of all seven games, there are exactly two pairs of games, the sum of whose runs was exactly 9 (a game can be used in multiple pairs).

Based on this information, which of these statements is accurate?
1. The Detroit Tigers won the World Series.
2. The Cincinnati Reds won the World Series.
3. The World Series winner is uncertain.

The bracketed clue numbers [1]-[7] are used for reference in the answer explanation.

The Detroit Tigers won the World Series.

[2]: Since there were seven games, this must be 0, 3, or 6 games. We know both teams scored an even number of runs in the whole series. This narrows it down from {0, 3, or 6} to 3. So each team had 3 even-run games, and 4 odd-run games.

[3]: The only case in which two whole numbers may be multiplied together to produce a prime number is that one number is 1 and the other is prime. Since one clue mentions that both teams had some prime-number-run games, the statement is shown to be merely an obfuscated way of saying "there were no games with exactly 1 run for either team."

[1] and [4]: These clues become very useful when combined, for no number exists that is both a perfect square and a prime number. So for each team, these make up 6 of the 7 games, leaving only one game for each team in which neither a prime number nor a perfect square of runs was scored. Here is the breakdown of the numbers 0 to 12, aside from the eliminated numbers 1 and 5:
Prime - 2,3,7,11 [4 games for each team]
Square - 0,4,9 [2 games each]
Neither prime nor square - 6,8,10,12 [1 game each]

Since all of the "neither prime nor square" numbers are even, this leaves 2 even and 4 odd games to make up the 6 "prime or square" games for each team. Since we know there was exactly one shutout, one team's square games were either {0,4} or {0,9}, and the other team's square games were either {4,4}, {4,9}, or {9,9}.

[7]: It can be found that the Reds must have one of the following subsets: {3,6,0,9}, {3,3,6}, {2,2,7}, {2,7,7}, {3,6,2,7}, {2,7,0,9}. {2,7,0,9} is the only one that can be completed legally: {2,7,0,9,3,3,12}.

We now know the Tigers' square games are either {4,4}, {4,9}, or {9,9}. {4,4} is the only one that can be continued legally, which forces their 4 primes to be odd. So Detroit did not score less than 3 in any game, which means the Reds could have won a maximum of just three games: those in which they scored 7, 9, 12.

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