The Lemon Fool

Three colleges are competing in an athletics competition. Susan, a student at Trent, attended to support her boyfriend, the college 100 metres champion. When she returned home, her father asked how her college had done. "We won the 100 metres all right," she replied, "but Derwent won the whole competition. They finished with 22 points, we got 9 and so did Mersey."

"How were the events scored?" asked her father.

"I can't remember,” replied Susan, "but there was a certain number of points for the winner of each event, a smaller number for second and a smaller number still for third. The numbers were the same for all events."

"How many events were there?"

"I don't know, dad. All I watched was the 100 metres."

"Was there a long jump?” asked Susan's brother. Susan nodded.

Which college won the long jump?

Cinelli

Does each team only have one competitor in each event?

John

cinelli wrote:Three colleges are competing in an athletics competition. Susan, a student at Trent, attended to support her boyfriend, the college 100 metres champion. When she returned home, her father asked how her college had done. "We won the 100 metres all right," she replied, "but Derwent won the whole competition. They finished with 22 points, we got 9 and so did Mersey."

"How were the events scored?" asked her father.

"I can't remember,” replied Susan, "but there was a certain number of points for the winner of each event, a smaller number for second and a smaller number still for third. The numbers were the same for all events."

"How many events were there?"

"I don't know, dad. All I watched was the 100 metres."

"Was there a long jump?” asked Susan's brother. Susan nodded.

Which college won the long jump?

Cinelli

If the problem is solvable then it must be Derwent:

If all three teams won an event there is not enough information to determine team which won the long jump. Therefore, at most two teams won an event. We know that Trent won one event, and Derwent got by far the highest points total. So they must have won the remaining events -- including the long jump.

I can't be bothered to work out the actual scores, though that was probably the point of the problem!


GS

GoSeigen wrote:If the problem is solvable then it must be Derwent: ...

Yes, that was the obvious answer, but the number of competitors is also relevant as asked earlier, whether or not all colleges competed in every event, if fractions of points can be earnt or lost, as well as the possibility of negative numbers or zero for being placed third (or second) ...

To put it another way, I failed miserably.

I don't have an answer but I know the following conclusions can be drawn:

Max points for a win is 9 (T won the 100m)
Min number of events is 3 (D scored 22, max for win is 9)

After that I have to start making assumptions - like points are whole numbers and positive.

Number of points for each event is F+S+T (First place + Second place + Third place)
Total points awarded is 22 + 9 + 9 = 40
F+S+T|40 (F+S+T divides 40)
we're also told F > S > T (I'm assuming also that T > 0)
so at a minimum F=3, S=2, T=1, but in this case 6 does not divide 40
So F+S+T is either 8, 10, 20 or 40 (5, 4, 2, or 1 events, respectively)

Trent won 100m and had total score of 9pts, so F =< 9
Derwent had 22pts, so there must have been more than 2 events to achieve that.
So either
F+S+T = 8 with 5 events or
F+S+T = 10 with 4 event

If it were 4 events (F+S+T = 10) then F >= 6 for Derwent to get 22 pts
since Trent won at least 1 event and got 9pts total, then F=6 and T=1, therefore S=3.
But if Derwent came second in 100m and won other 3, they would have (6x3)+3 = 21 points, which is a contradiction.

Therefore there must be 5 events, with F >=5 for Derwent to get 22 pts.
If F=5 then for Trent to get 9 points, F = 5 and T = 1, and Trent come 3rd in other 4 events. Therefore, also, S = 2
Therefore Derwent come second in 100m (2pts) and win other 4 events (4 x 5pts) to make 22 pts (therefore, including the long jump!)
Therefore Mersey comes third in 100m and second in every other event (1+2+2+2+2).

-Dom

psychodom wrote:Number of points for each event is F+S+T (First place + Second place + Third place)
Total points awarded is 22 + 9 + 9 = 40
F+S+T|40 (F+S+T divides 40)
we're also told F > S > T (I'm assuming also that T > 0)
so at a minimum F=3, S=2, T=1, but in this case 6 does not divide 40
So F+S+T is either 8, 10, 20 or 40 (5, 4, 2, or 1 events, respectively)

-Dom

Very good Dom!

Although, what if there are fewer than 3 competitors in an event? In that situation, with the scoring system you've deduced, it's possible Mersey could have won the long jump if they were the only competitor and there were 6 events and the other events had only 2 competitors. I think... You're no doubt correct... I'll shut up...

Or if there were more than 3 competitors so a college could score 1, 2 and 3 in an event. I believe then the problem becomes unsolvable since there are multiple combinations that would fit the data given. Which was why I asked the question.

John

Rereading the question, it does seem ambiguous, so sorry about that. But you don't want to set so many conditions that it looks like it should appear in an exam.

Dom has the solution I was looking for, so congratulations to him.

Cinelli