Say there are N gladiators, and I want to find the champion among them. The method by which this is done is a series of elimination rounds, which are similar to but not quite like the methods used in many modern sports events:(adsbygoogle = window.adsbygoogle || []).push({});

Each gladiator is given a number from 1 to N. Each gladiator then fights the n gladiators with numbers immediately lower than their own and the n gladiators with numbers immediately higher than their own, where n is some constant much smaller than N. For example, if n=2, then gladiator #50 fights gladiators #49, #48, #51, and #52 in the first round.

Only those who winallof their fights advance to the next round, and the process repeats. To continue the example, assuming that our #50 won all four fights, then all of their opponents were necessarily eliminated because they lost at least one of their fights, namely the one with #50. So, in the second round, #50's closest possible opponents would be #47, #44, #53, and #56. Then again, it's possible that they are already the "last man standing", because nobody else managed to win all of their four fights.

I haven't decided what happens "at the edges", i.e. who #1 fights in the first round, other than #2 and #3, if anyone. For practical reasons, they can't fight #N-1 and #N, which would of course be the tidiest solution. Let's treat that point as negligible, for now.

It's easy to calculate how many rounds this championship will take at least - one, see above. It's almost as easy to calculate how many rounds it will take at most. However, how many rounds will it taketypically, as a function of N and n?

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# Maths: Gladiatorial league

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