Suppose young Calvin has a collection of \(24\) toy cars that he wants to rank in terms of speed. He doesn't have a stopwatch, (or a radar gun); all he has is a track on which he can race a maximum of \(4\) cars against each other at a time.

Assuming that each car has a different speed and moves at that speed in every race, what is the minimum number of races Calvin will have to run in order to exhaustively rank his car collection according to speed?

(Note that we are looking for the minimum number of races that are necessary regardless of luck. In the best-case scenario we would need only \(9\) races. What we are looking for here is the minimum number of races in the worst-case scenario.)

This is a challenge question for those who enjoy optimizing sorts. I don't know if an achievable minimum is known; I believe that the theoretical minimum is \(18\), but I can't see this being achievable in reality. I know of an algorithm that has a minimum of \(28\) races, so the "challenge" is to devise a sort that can beat this value.

I don't know what kind of response to expect with this post, but it's just meant as an enjoyable diversion to play around with and to share what you know about optimization problems.

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