Two brothers, Bill and Phil, competed in a five-man nine-ball tournament. Each of the five players played one match of nine-ball against each of the others (and no ties). Each player received one point for each match he won; additionally, each player received one-half point for each match won by each opponent whom he defeated. Afterwards, each player put up equal money, such that the total pot in USD was equal to 10 times the total number of points won by all the players. Then the pot was redistributed back to the players according to their performance: each player received $10 for each point he earned during the competition. Bill beat his brother in their nine-ball match but ended up losing $11 on the night. How much did Phil win/lose on the night?
A bit, I don't if I had some luck in the thought process that got me there. I can't seem to be able to spoiler it...
[hidden]I realized that the maximum number of points was 200 (all win 2 and lose 2) and the minimum 150 (p1>p2>p3>p4). So with the variations in between we can have 150, 160,170,180,190,200 as the total pot, which gives 30,32,34,36,38,40 as possibilities for the payment required. So the total points of Bill are p=(payment+earnings) and with earnings = -11 the only possibility to get a multiple of 5 for p is if payment was 36. So Bill won 25, paid 36 and the pot was 180. When the total payout is 180, it means that we are 2 steps off the maximum result so we have 2 players with 3 wins, 1 with 2 wins and 2 with 1 win. The 25 can be possible then with one win against one player (Phil) who won his remaining 3 games (if it was two wins, one of the players must have won 1 and the other 0, which is not possible). Phil's wins must have come against the remaining players with 3,2 and 1 wins respectively. His total points are then 3*10+3*5+2*5 + 5 = 60, meaning his earning were 60-36 = 24[\hidden]
Palynka, very good. Maybe this variation below will be more challenging for you (sorry, sometimes it is difficult to figure out how challenging the question will be when I think it up).
This part is all the same as before:
Two brothers, Bill and Phil, competed in a five-man nine-ball tournament. Each of the five players played one match of nine-ball against each of the others (and no ties). Each player received one point for each match he won; additionally, each player received one-half point for each match won by each opponent whom he defeated. Afterwards, each player put up equal money, such that the total pot in USD was equal to 10 times the total number of points won by all the players. Then the pot was redistributed back to the players according to their performance: each player received $10 for each point he earned during the competition.
Now suppose after the tournament Bill and Phil had the following discussion:
Bill: "I cannot believe you won money on the night, whereas I did not. We both won the same number of matches!"
Phil: "Quit complaining. You know that's the way it goes with that format sometimes."
Bill: "If only Ray had beaten Jon in that last match, then we both would have made money on the night."
How much did Phil win on the night?
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Originally posted by LemonJelloI screwed up by missing "whom he defeated" in "each player received one-half point for each match won by each opponent whom he defeated." So I came up with a total of 30 points being awarded to the players no matter how they did. This made the 11 point difference impossible.
Palynka, you are correct. Was that too easy?