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Mathematics Forum
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| Author | Message / Information |
| alun Quote | Reply | | The Athletics Team Puzzle posted on: 8/18/2006 5:59:59 AM Assume I need to pick an athletics team. I have 6 athletes to choose from. All I have to base my decision on is their recorded times from each of their last 10 runs. I decide that a good way to decide who to pick is to create 'virtual races'. Let me explain. I will label each run numerically, so run 1 was their most recent time, and run 10 was the time from their least recent run. I will label each athlete alphabetically. So A1 means the recorded time from athlete A's most recent run. I've already calculated that overall this will produce 1,000,000 virtual races (6^10). So for instance one of those virtual races will be A1,B1,C1,D1,E1,F1. Another might be A1,B1,C1,D1,E1,F2. And so on until all 1,000,000 possibilities are covered. What I want to know is this. Is there a way I can find out how many of these virtual races that each athlete has won. The only way I can think of is to create each one of the 1 million virtual races one by one, but there must be a better way I think. What I've done so far is work out how many virtual races this would create ie 6^10 if assessing 6 athletes. I've also figured out that the fastest time of all 60 times in front of me, must necessarily win every single virtual race that it is involved in (though I don't know how many that would be either!) Nonetheless, I've realised that one can take the fastest time and say 'right, this time would have been enough to win x races'. As such, I'm hoping maybe that by sorting the times in order (best to worse) that I can simply assign race wins to each of the 60 times rather than creating each of the 1 million virtual races individually. Many thanks for any help on this chewy problem Al |
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Euler
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The Athletics Team Puzzle
replied on: 8/20/2006 10:30:10 AM Well, from the way you describe the problem, I think that there are only 10 (one for each of the ten races) possibilities- not 6^10. I say this because you have the time labeled as the most recent to the least recent, so if these athletes were running together, then the possiblilities are A1,B1,C1,D1,E1,F1, or A2,B2,C2,D2,E2,F2 and so forth. This is because it makes no sense (to use your example of A1,B1,C1,D1,E1,F2) for the most recent times of athletes A-E to occur with the second most recent time of athletes. IF this is the case, and there are only six possibilities, and you have the times of each runner for each race- you can easily determine the finish positions of each runner in each race and compare them to decide which runners you want- however many that may be. |
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