Tag Archives: recreational mathematics

Perfect Partitions

I was playing with a variant of my Containers of Water question where we were instead interested in solid weights on a scale. It occurred to me that, as is often the case, I should consider easier problems first and see what comes of them. Ultimately, this led me down a path towards the idea of a ‘perfect partition’ and a few papers published in the late 1800s by MacMahon. Here’s how that went:


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Containers of Water II

In a previous post, I considered the following two questions:

What sets $latex S $ maximize $latex |{\bf F}(S;p)| $ for various $latex p$?
What sets $latex S $ maximize $latex \lfloor {\bf F}(S; p) \rfloor $ for various $latex p$?

I then changed the first question, which I think is perhaps not so interesting, to the following:

What sets S maximize |{\bf F}(S;p)|_c, where |\cdot|_c denotes the largest connected interval of results?

Let’s explore a few cases to see what these answers might look like.


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