Eight athletes took part in a recent sporting competition which comprised several events. The final standings were as follows:
Position Points
1 2
2 18
3 40
4 54
5 70
6 112
7 240
8 448
How many events were there in the competition and how were the points awarded?
Cinelli
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Sport

 Lemon Quarter
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Re: Sport
The question is underspecified.
Looks as if it may embody some assumptions about sporting competitions. I thought they awarded medals, rather than points?
Oh, right, I've got it. A single hole at golf. The #1 player made it in two shots; others struggled a bit.
Looks as if it may embody some assumptions about sporting competitions. I thought they awarded medals, rather than points?
Oh, right, I've got it. A single hole at golf. The #1 player made it in two shots; others struggled a bit.

 Lemon Quarter
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Re: Sport
cinelli wrote:Eight athletes took part in a recent sporting competition which comprised several events. The final standings were as follows:
Position Points
1 2
2 18
3 40
4 54
5 70
6 112
7 240
8 448
How many events were there in the competition and how were the points awarded?
Something missing there, I think, otherwise there are huge numbers of solutions, such as:
* 984 events, each event awarded 1 point to the winner and none to anyone else, each athlete won the indicated number of events.
* 492 events, each event awarded 2 points to the winner and none to anyone else, each athlete won half the indicated number of events.
* 2 events, awarding 224, 120, 56, 35, 27, 20, 9 and 1 points for 1st8th places, the athletes came in the same order in both events.
Of course, 984 and 492 are rather larger than what people tend to mean by "several", and 2 rather smaller, so the intended answer might involve the number of events being in the right range to count as "several". But different people have different ideas about just what that range is, so if that's the case, an explicit range ought to be stated...
Other information that might be missing includes whether each event awarded points in the same way, whether points only depend on the athlete's position or depend in some way on their exact performance (competitions such as the heptathlon do the latter), and whether points from different events are combined by adding them together or might be combined in some other fashion (e.g. by multiplying them together).
Gengulphus

 Lemon Quarter
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Re: Sport
cinelli wrote:Eight athletes took part in a recent sporting competition which comprised several events. The final standings were as follows:
Position Points
1 2
2 18
3 40
4 54
5 70
6 112
7 240
8 448
How many events were there in the competition and how were the points awarded?
Possible spoiler of the intended solution, inspired by observing an otherwiseunlikely numerical property of the listed numbers of points...
The product of the listed numbers of points is 2^21 * 3^7 * 5^3 * 7^3, so is an exact cube. And its cube root is 2^7 * 3^2 * 5 * 7 = 1 * 2 * 3 * 2^2 * 5 * (2*3) * 7 * 2^3 = 1*2*3*4*5*6*7*8. So I suspect the intended solution is that there were three events, each scored purely on position and not performance, with awards from the three being multiplied together to produce each athlete's final score, and points for each event being awarded on a 1/2/3/4/5/6/7/8 basis, probably for 8th1st places in that order (but just possibly for 1st8th places, with the objective of the competition being to score as few points as possible).
But it's definitely not deducible from the information in the puzzle  it's just one possible solution from the many available that looks especially elegant, elegant enough to be likely to be the intended one.
Incidentally, three is for me "a few" rather than "several", which is a bit of a blemish on that solution to my mind  but I know not everyone feels that way.
Gengulphus

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Re: Sport
When my sister was very young I told her that several meant 'between 3 and 15'. For some reason it's one of those childhood memories that keeps being brought up. Anyway the dictionary says 'more than 2, but not many', so maybe I was pretty close.
In terms of sports I'm aware of biathlon, triathlon, pent.., hep,... and decathlon, but nothing more than 10.
In terms of sports I'm aware of biathlon, triathlon, pent.., hep,... and decathlon, but nothing more than 10.

 Lemon Quarter
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Re: Sport
It's occurred to me that the question about how the points are awarded might be intended to ask not just the basis on which they are awarded, but also what points each athlete was awarded, so here's a further spoiler about that on the assumption that the solution is the one in my last post that I think is intended:
Calling the athletes who were in positions 18 according to the table AH respectively, the three awards of 7 for second place must have gone to athletes E (score 70), F (score 112) and H (score 448), as those are the only three available factors of 7. Similarly, there are only three available factors of 6 from third place, from athletes B (score 18), D (score 54) and G (score 240), and the three available factors of 5 must come from athletes C (score 40), E (score 70) and G (score 240). Once the three third places are taken into account, the scores of athletes B, D and H's remaining events are 18/3 = 6, 54/3 = 18 and 240/3 = 80 respectively, and the first two of those are the only remaining sources of factors of 3 and they only provide three of them, so one of the three awards of 3 points for sixth place must have gone to B and the remaining two to D.
At this point, we've fully accounted for athlete D's score (a 3rd place and two 6th places for a score of 6*3*3 = 54). We've also got two of the positions for each of athletes B, E and G, from which we can deduce their remaining position. For B, it must score 18/(6*3) = 1, so B's remaining award is for an eighth place. For E, it must score 70/(7*5) = 2, so E's remaining award is for a seventh place. For G, it must score 240/(6*5) = 8, so G's remaining award is for a first place.
We've still to complete the sets of awards for athletes A, C, F and H, which have scores yet to be accounted for of 2, 40/5 = 8, 112/7 = 16 and 448/7 = 64 respectively. We have two awards of 8 for first place, three of 4 for fifth place, two of 2 for seventh place and two of 1 for eighth place yet to be assigned. Athlete H has two awards outstanding that multiply to 64, so they must be the two remaining awards of 8, which completes the set of first places. Then athlete F has two awards outstanding that multiply to 16, so they must be two awards of 4 for fifth place. The remaining award of 4 for fifth place must go to athlete C, then the two remaining awards of 2 must go one each to athletes A and C, and the two remaining awards of 1 for eighth place must go to athlete A.
So the positions that each athlete achieved and their score calculations are:
Can we identify each athlete's positions in the three events? Obviously not entirely, since we're not told anything about the order of the three  so we can swap the orders for the three events around as we like. I.e. any solution to that question has 3! = 6 trivial variations caused by permuting the events. However, we can fix on one of those by choosing to label the events according to the positions achieved by any particular athlete who achieved three different positions. So somewhat arbitrarily choosing that athlete, let events X, Y and Z be the events in which G achieved 1st, 3rd and 4th positions. Then we can immediately deduce that athlete H achieved 2nd place in X and 1st places in both Y and Z, so what we know about the individual event outcomes so far is:
Athlete F must have achieved 5th place in event X, since 2nd place is already taken, and similarly athlete D must have achieved 6th place in event Y.
Athletes E and F must have achieved the two remaining 2nd places, which are in events Y and Z. Suppose E did so in event Y and F in event Z. Then F's remaining 5th place must have been in event Y and so C's 5th place is in event Z, and then looking at the remaining 4th places, E's must have been in event X and C's in event Y. Alternatively, F did so in event Y and E in event Z, and then it follows that the remaining 5th places went to C in event Y and F in event Z and the remaining 4th places to C in event X and E in event Y. So we've got two possibilities:
Now look at the 7th places, which go to athletes A, C and E. In both cases, we already know that C and E are in other places in event Y, so A must be in 7th place in event Y and therefore must be in 8th place in events X and Z. In each case, the 7th places for C and E are determined by the fact that each of them has already been placed in one of events X and Z, so it must be in the other. So the two possibilities become:
Now we need to know which of B and D placed 3rd and which 6th in event X, and similarly in event Z, and all we know is that they're in the opposite orders in the two events. So each of those two possibilities leads to two possible final solutions.
So no, we cannot identify the exact outcomes of the three events  we can only narrow it down to one of four possibilities.
Gengulphus
Calling the athletes who were in positions 18 according to the table AH respectively, the three awards of 7 for second place must have gone to athletes E (score 70), F (score 112) and H (score 448), as those are the only three available factors of 7. Similarly, there are only three available factors of 6 from third place, from athletes B (score 18), D (score 54) and G (score 240), and the three available factors of 5 must come from athletes C (score 40), E (score 70) and G (score 240). Once the three third places are taken into account, the scores of athletes B, D and H's remaining events are 18/3 = 6, 54/3 = 18 and 240/3 = 80 respectively, and the first two of those are the only remaining sources of factors of 3 and they only provide three of them, so one of the three awards of 3 points for sixth place must have gone to B and the remaining two to D.
At this point, we've fully accounted for athlete D's score (a 3rd place and two 6th places for a score of 6*3*3 = 54). We've also got two of the positions for each of athletes B, E and G, from which we can deduce their remaining position. For B, it must score 18/(6*3) = 1, so B's remaining award is for an eighth place. For E, it must score 70/(7*5) = 2, so E's remaining award is for a seventh place. For G, it must score 240/(6*5) = 8, so G's remaining award is for a first place.
We've still to complete the sets of awards for athletes A, C, F and H, which have scores yet to be accounted for of 2, 40/5 = 8, 112/7 = 16 and 448/7 = 64 respectively. We have two awards of 8 for first place, three of 4 for fifth place, two of 2 for seventh place and two of 1 for eighth place yet to be assigned. Athlete H has two awards outstanding that multiply to 64, so they must be the two remaining awards of 8, which completes the set of first places. Then athlete F has two awards outstanding that multiply to 16, so they must be two awards of 4 for fifth place. The remaining award of 4 for fifth place must go to athlete C, then the two remaining awards of 2 must go one each to athletes A and C, and the two remaining awards of 1 for eighth place must go to athlete A.
So the positions that each athlete achieved and their score calculations are:
Athlete 1st 2nd 3rd 4th 5th 6th 7th 8th Score

A       1 2 2*1*1 = 2
B   1   1  1 6*3*1 = 18
C    1 1  1  5*4*2 = 40
D   1   2   6*3*3 = 54
E  1  1   1  7*5*2 = 70
F  1   2    7*4*4 = 112
G 1  1 1     8*6*5 = 240
H 2 1       8*8*7 = 448

Can we identify each athlete's positions in the three events? Obviously not entirely, since we're not told anything about the order of the three  so we can swap the orders for the three events around as we like. I.e. any solution to that question has 3! = 6 trivial variations caused by permuting the events. However, we can fix on one of those by choosing to label the events according to the positions achieved by any particular athlete who achieved three different positions. So somewhat arbitrarily choosing that athlete, let events X, Y and Z be the events in which G achieved 1st, 3rd and 4th positions. Then we can immediately deduce that athlete H achieved 2nd place in X and 1st places in both Y and Z, so what we know about the individual event outcomes so far is:
Event 1st 2nd 3rd 4th 5th 6th 7th 8th

X G H      
Y H  G     
Z H   G    

Athlete F must have achieved 5th place in event X, since 2nd place is already taken, and similarly athlete D must have achieved 6th place in event Y.
Event 1st 2nd 3rd 4th 5th 6th 7th 8th

X G H   F   
Y H  G   D  
Z H   G    

Athletes E and F must have achieved the two remaining 2nd places, which are in events Y and Z. Suppose E did so in event Y and F in event Z. Then F's remaining 5th place must have been in event Y and so C's 5th place is in event Z, and then looking at the remaining 4th places, E's must have been in event X and C's in event Y. Alternatively, F did so in event Y and E in event Z, and then it follows that the remaining 5th places went to C in event Y and F in event Z and the remaining 4th places to C in event X and E in event Y. So we've got two possibilities:
Event 1st 2nd 3rd 4th 5th 6th 7th 8th

X G H  E F   
Y H E G C F D  
Z H F  G C   

Event 1st 2nd 3rd 4th 5th 6th 7th 8th

X G H  C F   
Y H F G E C D  
Z H E  G F   

Now look at the 7th places, which go to athletes A, C and E. In both cases, we already know that C and E are in other places in event Y, so A must be in 7th place in event Y and therefore must be in 8th place in events X and Z. In each case, the 7th places for C and E are determined by the fact that each of them has already been placed in one of events X and Z, so it must be in the other. So the two possibilities become:
Event 1st 2nd 3rd 4th 5th 6th 7th 8th

X G H  E F  C A
Y H E G C F D A B
Z H F  G C  E A

Event 1st 2nd 3rd 4th 5th 6th 7th 8th

X G H  C F  E A
Y H F G E C D A B
Z H E  G F  C A

Now we need to know which of B and D placed 3rd and which 6th in event X, and similarly in event Z, and all we know is that they're in the opposite orders in the two events. So each of those two possibilities leads to two possible final solutions.
So no, we cannot identify the exact outcomes of the three events  we can only narrow it down to one of four possibilities.
Gengulphus

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Re: Sport
Reply to spoiler:
Very well solved, Gengulphus. Yes, I admit the wording of the puzzle was a little woolly but deciding what was intended was part of the challenge. Did you think that I made up the whole idea of multiplying positions to establish the winner of of some kind of combined competition? No. In August I saw on TV a climbing competition held in Japan. Climbing up an artificial wall, that is. There were three kinds of climbing: “lead” where they ascend the wall attaching their safety rope to clips as they rise; “speed” where the climbers sprint up a ten metre wall in under ten seconds; and “bouldering” where they cling on to tiny finger holds, often upside down, as they reach a goal. The eight climbers, men and women, are placed in order for each event – 1 to 8  and the positions attained multiplied together to obtain a combined ranking. There might have been medals awarded for each discipline but there were certainly gold, silver and bronze for the top three in the combined competition. I found the whole affair fascinating. Moreover next year it will be an Olympic sport. The Japanese are very good at climbing and they will do well with home advantage.
The gold medal winner came first, first and second in the three disciplines (combined product 2). The man who came last (combined product 448) came seventh in event one but injured a finger and dropped out, being graded as last in the other two events. Injuring a finger is very serious for a climber as you can imagine the stress hanging upside down places on the digits.
Anyway that was the basis of the puzzle. I had to change the positions very slightly to make an unambiguous solution. Note that the combined order would have been the same if the positions had been added rather than multiplied.
Cinelli
Very well solved, Gengulphus. Yes, I admit the wording of the puzzle was a little woolly but deciding what was intended was part of the challenge. Did you think that I made up the whole idea of multiplying positions to establish the winner of of some kind of combined competition? No. In August I saw on TV a climbing competition held in Japan. Climbing up an artificial wall, that is. There were three kinds of climbing: “lead” where they ascend the wall attaching their safety rope to clips as they rise; “speed” where the climbers sprint up a ten metre wall in under ten seconds; and “bouldering” where they cling on to tiny finger holds, often upside down, as they reach a goal. The eight climbers, men and women, are placed in order for each event – 1 to 8  and the positions attained multiplied together to obtain a combined ranking. There might have been medals awarded for each discipline but there were certainly gold, silver and bronze for the top three in the combined competition. I found the whole affair fascinating. Moreover next year it will be an Olympic sport. The Japanese are very good at climbing and they will do well with home advantage.
The gold medal winner came first, first and second in the three disciplines (combined product 2). The man who came last (combined product 448) came seventh in event one but injured a finger and dropped out, being graded as last in the other two events. Injuring a finger is very serious for a climber as you can imagine the stress hanging upside down places on the digits.
Anyway that was the basis of the puzzle. I had to change the positions very slightly to make an unambiguous solution. Note that the combined order would have been the same if the positions had been added rather than multiplied.
Cinelli

 Lemon Quarter
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Re: Sport
StepOne wrote:When my sister was very young I told her that several meant 'between 3 and 15'. ...
How helpful of you  considering that there are two widespread ideas about what "between" means! ;)
Gengulphus
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