Ranking And Average

Question

Four athletes A, B, C and D are evaluated on four different parameters – S, A, I and T. Further it is known that:

I. Each athlete was given a score, which was an integer between 4 and 10 (both inclusive) on every parameter.

II. The sequence of the athletes, when arranged in descending order of the scores received in S was A, B, C and D.

III. The sequence of the athletes, when arranged in descending order of the scores received in A was B, D, C and A.

IV. The sequence of the athletes, when arranged in descending order of the scores received in I was C, A, D and B.

V. The sequence of the athletes, when arranged in descending order of the scores received in T was D, B, A and C.

VI. A did not get 9 in any of the parameters while C got 9 in exactly one parameter.

VII. The average scores of the four athletes in the parameters S, A, I and T were 7, 8, 6 and 8 respectively.

VIII. Every athlete had different scores for each of the four parameters. Similarly, each athlete had a different score from the other athletes on any of the four parameters.

Please tell how to solve this. Should I solve kakuro to get better at such puzzles?

Answer 1

Okay, there’s a lot to tackle here, but let’s see what we can do:

A was the highest scorer in S, but did not score 9. If they scored 8, then all would have to be different, so the scores in S would be 8,7,6,5 — an average of 6.5 instead of 7. Therefore A scored 10 in S. Similarly, C was the highest scorer in I but they were 3rd ranked or worse in each of the other categories (meaning in each of the other categories, their score is maximum 8). Therefore C scored a 9 in I. We know the average for I is 6, so the others (without C) scored an average of 5. (Note 6 x 4 = 24 total points, and without C the other three must have scored 15 total points). Since all must have scored differently, the only way to do this with a minimum score of 4 is 6-5-4. Therefore for I: A-B-C-D was 6-4-9-5. There are two ways to average 8 for four people with max score 10 and no repetition: 10-9-7-6 and 10-9-8-5. Then for A, if B-D-C-A was 10-9-7-6 — contradiction, because A scored 6 in I. Therefore for A, A-B-C-D scored 5-10-8-9. For T, A-B-C-D could score either 8-9-5-10 or 7-9-6-10. We’ll come back to this. For S, A scored 10, the others must have averaged 6. B came second but couldn’t score 9 because they’ve already done so in T. So let B score 8. Then the other two must have averaged 5 together. The only possible way is for A-B-C-D to score 10-8-6-4. So C scored 6 in S, so C can’t score 6 in T. Therefore the T scores are 8-9-5-10.

So the final grid is

_|A|B|C|D
S|10|8|6|4
A|5|10|8|9
I|6|4|9|5
T|8|9|5|10

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Edit: Advice section for this puzzle:

The above was kind of stream of consciousness for how I solved this problem. While process of elimination was important, I found the most important clues were VII and VIII. (I suppose they were all equally important — can’t figure out who scored what without the order of the scoring, etc., but I digress). Ensuring that no two athletes scored the same in any of the parameters made the averages much more useful, since there are a limited number of ways to sum to a number (eg. the average x 4, or any of the adjusted averages x 3 or x 2) while using different numbers between 4 and 10. Forcing each athlete to have different scores for each parameter brought me the rest of the way.

Answer 2

Providing a rudimentary guess at this ...

First guess i would take is to come out with a basic set of values for each parameter. Since as per VIII (No two athletes scored equal on any of the four parameter) and that number of athletes are even, the initial guess would be Average-2,Average-2,Average+1,Average+2


_|A|B|C|D
S|9|8|6|5
A|6|10|7|9
I|7|4|8|5
T|6|9|7|10



Solving for VI, A cannot have a score of 9 in S and C has to have a score of 9 in I (changes in any other variable would have to shift all other variables, affecting the Parameter Average.



For S, I could choose between B & D to decrease by 1. D appears to be a good choice as it resolves another conflicting issue with parameter I (VIII, no athletes scored equal on 2 parameters)
For I, since D cannot be reduced by 1 post-the earlier decision to bring S down to 4, the choice remains between A & B. A is out of the option as reducing 7 to 6 increases the repeated scores for A.



Remaining scores to deconflict include Parameter A & I for Athlete A/C. Increasing either parameter score for C to 8 and decreasing the corresponding score of Athlete A to 5 will solve the problem.

Final grid:

_|A|B|C|D
S|10|8|6|4
A|5|10|8|9
I|7|3|9|5
T|6|9|7|10