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Necessary information. The sport climbing consists of three disciplines: speed climbing, bouldering, and lead climbing. In combined events, the disciplines come in this order, and the combined score of an athlete is a product of his/her standings in all three disciplines—with the lowest score ranking higher. In case if equality of the scores, the qualifications standings are compared.

Say, athlete A finished 2nd in speed, 5th in bouldering, and 6th in lead, athlete B, 4th in speed, 3rd in bouldering, and 5th in lead, athlete C, 3rd in speed, 6th in bouldering, 4th in lead. Then the combined score of A is $2\times 5\times 6 = 60$, the score of B is $4\times 3\times 5 = 60$, and the score of C is $3\times 6\times 4 = 72$. So the athlete C will rank lower than A and B, and among those two, the one who ranked higher in the qualification for the event will rank higher in the event as well.

Now here is the puzzle (assume that there can never be ties in any discipline).

DISCLAIMER The story, all names, characters, and incidents portrayed in this post are fictitious. No identification with actual persons (living or deceased) is intended or should be inferred.

Eight athletes—Adam, Bessa, Colin, Daniel, Fedir, Guillaume, Hannes, and Jacob—take part in a climbing combined final; their qualification stangings are in the reverse alphabetical order. All athletes have already finished all their attempts with only Jacob left to climb the lead route. It appears that:

  • each of the athletes may get a medal (rank first, second or third) in the combined event, depending on Jacob's rank in the lead climbing;
  • none of the atheles is guaranteed to get a medal;
  • the eight possible medal-stand trios are different.

What are the atheles' ranks in all three disciplines?

I hope the question is clear (please feel free to ask if it's not). So far I managed to find one solution, but there might be more.

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    $\begingroup$ "their qualification stangings are in the reverse alphabetical order." Not quite sure what you mean here? $\endgroup$
    – acrabb3
    Commented Aug 6, 2021 at 12:27
  • $\begingroup$ @acrabb3, this is needed to determine who ranks higher in the case of equality of combined scores. The ranking is: A<B<C<D<F<G<H<J. It is actually not needed to solve the problem. I though this would exclude extra solutions obtained by permutation of athletes, but it wouldn't, so this part of formulation is not really important. $\endgroup$
    – zhoraster
    Commented Aug 6, 2021 at 12:34
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    $\begingroup$ Love the disclaimer :) $\endgroup$ Commented Aug 6, 2021 at 14:59

2 Answers 2

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Here is a solution found using Z3 (python code here):

Athlete Speed Ranking Bouldering Ranking Lead Ranking
Adam 8 6 1
Bessa 6 4 2
Colin 3 5 3
Daniel 5 2 4
Fedir 1 8 5
Guillaume 7 1 6
Hannes 2 3 7
Jacob 4 7 -

And here are the possible scores, with the medal-winners highlighted:

Jacob's Rank Adam Bessa Colin Daniel Fedir Guillaume Hannes Jacob
1 96 72 60 50 48 49 48 28
2 48 72 60 50 48 49 48 56
3 48 48 60 50 48 49 48 84
4 48 48 45 50 48 49 48 112
5 48 48 45 40 48 49 48 140
6 48 48 45 40 40 49 48 168
7 48 48 45 40 40 42 48 196
8 48 48 45 40 40 42 42 224
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Here is my solution (found manually):

Athlete Speed Ranking Bouldering Ranking Lead Ranking
Adam 2 4 5
Bessa 8 3 2
Colin 3 5 3
Daniel 7 1 6
Fedir 1 6 7
Guillaume 6 8 1
Hannes 5 2 4
Jacob 4 7 -

The final scores:

Jacob's Rank Adam Bessa Colin Daniel Fedir Guillaume Hannes Jacob
1 48 72 60 49 48 96 50 28
2 48 72 60 49 48 48 50 56
3 48 48 60 49 48 48 50 84
4 48 48 45 49 48 48 50 112
5 48 48 45 49 48 48 40 140
6 40 48 45 49 48 48 40 168
7 40 48 45 42 48 48 40 196
8 40 48 45 42 42 48 40 224

What is interesting is that while the solution is essentially different from @2012rcampion's, the second table is the same (up to permutation of athletes).

My solution has an additional peculiarity (which is also possible to achieve with @2012rcampion's solution by permuting athletes):

  • One of the athletes except Jacob (here, Guillaume) may place first and last.
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    $\begingroup$ Well done. I see Jacob first and last as well. The second table does not match 2012campion's. For example, his Fedir wins a medal in seven cases while you have nobody who does that. $\endgroup$ Commented Aug 10, 2021 at 15:27
  • $\begingroup$ @RossMillikan, oh yes, Jacob, didn't think about him :D Concerning Fedir winning 7 medals, this may be fixed by renaming athletes or changing the qualification ranks (though nobody in Ukraine would be willing to "fix" that). $\endgroup$
    – zhoraster
    Commented Aug 10, 2021 at 20:04

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