# Who Scored How Many Points

Lalo, Tyson, and Michael played a number of games of pick-up basketball. At the end, their combined points across all the games were:

• Lalo: 20
• Tyson: 10
• Michael: 9

They noticed that in every game, one of them scored x points, one of them scored y points, and one of them scored z points, where x > y > z and all three are distinct positive integers.

If Tyson got the highest score in the first game, who got the second highest score in the second game?

#### Click to show solution

Michael got the second highest score in the second game.

The total points scored between them is 39, and since the three scores are distinct integers, they must have scored at least 1 + 2 + 3 = 6 points combined each game. The only possible game counts are 1, 3, 13, and 39 (the prime factors of 39), but we can rule out 13 and 39 because the total score would exceed 39 and we can rule out 1 because they played at least 3 games. So they must have played 3 games, scoring a combined x + y + z = 13 points each game.

Since Tyson got the highest score in the first game (x) but only 10 points overall, he must have scored x + z + z (if he had any other scores higher than z, then his total would be greater than or equal to x + y + z = 13).

Since Tyson scored z twice, then Michael must have scored at least y + y + z points. Michael couldn’t have scored higher than that, because then he would have scored at least 13 points. Since Tyson scored z on the second and third games, that means Michael must have scored y, the second highest score, in the second game.