Local battle
In rugby, points are scored in the following manner:
- Successfully completing a penalty kick is worth 3 points.
- Successfully completing a try is worth 5 points. After a try, a conversion kick is attempted: if successful, it is worth an additional 2 points.
In a recent rugby match between Berkeley's Golden Bears and the St. Mary's Gaels, St. Mary's led 17-11 at halftime. Berkeley stormed back in the second half to win 27-20.
For the entire game, Berkeley had 4 penalty kicks chances. How many tries did they score in the second half?
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1 solution
Let's start by figuring out Cal's first half scoring. Scores are either 3 points (penalty kick), 5 points (try without conversion), or 7 points (try with conversion.)
Cal scored 11 points. Since this is of the form 3n + 2, they must have either scored 5 points 3A + 1 times, or 7 points 3B + 2 times, plus some number of 3 point penalty kicks. They can't have scored any 7 pointers because the minimum number is 2, which would yield 14 points. Hence they scored 1 5 point try and 2 penalty kicks.
In the second half, they scored 16 points. Since all scores are an odd number of points (3, 5 and 7 are all odd), they must have scored an even number of times. They can't have only scored 2 times, because this would yield a maximum of 14 points. They can't have scored as many as 6 times, because this would yield a minimum of 18 points. Hence they scored exactly 4 times.
There are two possibilities for 4 scores adding to 16 points. 3 penalty kicks (9 points) plus one 7 point try (with conversion), or 2 penalty kicks (6 points) plus 2 5 point tries (without conversion.) But the first one yields a total of 2 + 3 = 5 penalty kicks for the game. So they must have scored 2 tries in the second half.