I'm asked to determine the best and worst final result of a series of 8 car races. One condition is that I will always finish 3rd. The points you get for each rank are listed in the following table:
The problem does not state what's meant by best and worst final result, but I'm thinking the best result is being the winner of the competition with the maximum point difference to the second contestant. I think I've already solved this one using R. race 2 is missing, but that's just a minor programming mistake.
I've calculated this by generating all possible combinations of final positions for a race and then kept only those where I finish 3rd. For the first race, I've selected a random result because the first one doesn't matter. for every following race I took the one which lead to the maximum point difference from me to the player with the highest points excluding myself.
I'm having trouble finding the worst possible result. I've changed the algorithm so that it takes the race result where i go down one rank. If that's not possible it takes the race result where the point difference from me to the opponent below me is the smallest. I've got some results this way but I'm not sure if they're the worst possible.
I'd like to ask if there is another approach to this problem, which isn't as "brute-force"-y as my solution, and I'd also like to know if I've really gotten the worst result, because I feel like I could lose another rank. The top-player is 16 points ahead of me and I feel like at least 13 of these points could be redistributed.
By analyzing the points available to the opponents we can establish the largest winning margin and the lowest finishing position.
There are $29$ points available in each race. You will get $5$ points, so the other drivers collectively get $24$ points per race. Across $8$ races this is $(8\cdot24)/7 \approx27.43$ each. You should be able to get the other drivers all to finish with either $27$ or $28$ points. You will win by $12$ clear points.
First, second, fourth and fifth places award $8+6+4+3=21$ points. Across the $8$ races, let four opponents finish in each of these positions twice. Each of those four opponents will have $42^{[1]}$ points. You will have $40$ points, so you will finish fifth.
Note that we can't extend this to a fifth opponent. There aren't enough points available to the opponents for you to finish sixth.
$^{[1]}$ This, of course, is the answer to everything.