Solution: Interfaculty championship

We publish the solution to the divertimento Interfaculty Championship.

The fun:

Three faculties, Mathematics, Physics and Chemistry, participated in a university sports championship. Each had one participant in each event.

Isabel, a student from the Faculty of Physics, sat in the stands to cheer on a friend of hers who was the swimming champion of her faculty. When she got home, her brother asked her how her faculty had done. “We won the swimming event,” she replied, “but Mathematics won the championship. They scored a total of 22 points, while Chemistry and we were tied at 9”.

“How do the tests score?” asked the brother. “I don’t know very well,” she said, “but there was a certain number of points for the winner of each test, a smaller number of points for the second and an even smaller number of points for the third. What I do remember,” she continued, “is that the score was the same for all the trials.”

“How many tests were there in all?” the brother asked her again. “I don’t know,” she replied, “the only event I was watching was the swimming competition.”

“Wasn’t there a high jump?” the brother asked again. “Yes, I saw it on a board.”

“And do you know who won it?” the brother asked finally. “No,” she replied, “I don’t know.”

Can you deduce, from this dialogue, the answer to this last question and, of course, the scores of all the championship events?

The solution:

The winner in the high jump was the faculty of Mathematics.

As the number of points assigned to the first, second and third participants in each event is a whole number, the score assigned to the winner of each event cannot be less than 3.

It is also known that the championship had two or more events and that the faculty of Physics was the winner in Swimming. Since the final score of that faculty was 9, the number of points for the winner of a race cannot be more than 8, as it is known that there were at least 2 competitions.

But 8 cannot be that number, because in that case there could only be two tests and then the faculty of Mathematics could not accumulate 22 points. Nor can it be 7, because then there could not have been more than 3 tests due to the final score of Physics and then Mathematics could not have scored 22 points either.

Similar reasonings eliminate the possibility of the first place in each test being scored 4 or 3 points. Therefore, by exclusion, the only possible score for the winner has to be 5 or 6 points. This last option can be discarded with a more detailed argument in the vein of the following one, so let us focus on the case of 5 points awarded to the faculty in the first place.

In this case, the championship had to consist of 5 events, for less than this would not allow the faculty of Mathematics to reach 22 points and more would increase the total of Physics, the winning faculty in swimming, which scored only 9 points.

Consequently, the other four scores of this school had to be 1 point each. So the faculty of Mathematics now has only two ways to reach 22 points: 4, 5, 5, 5, 5, 5, 3 or 2, 5, 5, 5, 5, 5, 5.

However, the first possibility has to be eliminated because it needs four different scores (and would give the faculty of Chemistry a total of 17 points and it is already known that it only had 9). The only remaining possibility gives the faculty a total of 17 points, so the final result of scores and competitions is as follows:

Faculty	P1	P2	P3	P4	P5	TOTAL
Mathematics	2	5	5	5	5	22
Physics	5	1	1	1	1	9
Chemistry	1	2	2	2	2	9