The Mathlon

From the 1986 Canadian Maths Olympiad:

A, B and C are the participants in a Mathlon – an athletics competition made up of several events. In each event x points are awarded for finishing 1st, y points for finishing 2nd, and z points for finishing 3rd, where x>y>z are positive integers. In the end, A scored 22 points, and B and C each scored 9 points. B won the 100 metres. How many events were there in total, and who finished second in the high jump?

I was immediately intrigued by this problem the first time I read it; on the face of it, it doesn’t seem like we have enough information to solve it. Of course, that’s where all the fun is…

Okay, 40 points are handed out in total, so the number of points per event must be a factor of 40. Since x\geq 3, y\geq 2 and z\geq 1, the number of points per event is at least 6. This leaves 8, 10 and 20 as our only options (we cannot have 40 points per event since we know there were at least two events). So the number of different events is 5, 4 or 2.

Suppose there were only 2 events, with 20 points per event. 7 points to a winner leaves at most 6 points for 2nd and at most 5 for 3rd, which is less than 20 in total. So the winner of an event must receive at least 8 points. But B won an event, yet only ended up with 9 points in total. Thus a win must be worth exactly 8 points. So there is no way A could amass 22 points with just two events.

Suppose there were only 4 events, with 10 points per event. B won an event and ended on 9 points, so in this case a win must be worth at most 6 points. This leaves possible points per event as (6,3,1), (5,4,1) or (5,3,2). But again, this makes it impossible for A to amass 22 points.

Therefore, there must have been exactly five events, with 8 points per event. B won an event and ended on 9 points, so the winner must receive at most 5 points. This leaves possible points per event as (5,2,1) or (4,3,1). Since A accumulated 22 points, the points distribution per event can only be (5,2,1). So A finished 2nd in the 100 metres, and won every other event. B won the 100 metres, and came last in every other event. Hence C finished last in the 100 metres, and finished 2nd in every other event, including the high jump.

I actually solved this problem with a friend in a restaurant; we managed to talk it through without having to write anything down, which can’t be said for lots of interesting puzzles. This is definitely one I will be sharing with other keen problem solvers.

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