# 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:

Posted in Expository, Math.CO, Math.REC, Mathematics | | 2 Comments

## Containers of Water II

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

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

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.

Posted in Math.REC, Mathematics, Open | | 1 Comment