# Group Theory in the Bedroom

I had never thought of this:

In a sense, base 3 is the best of the integer bases because 3 is the integer closest to

e…Suppose you are creating one of those dreaded telephone menu systems — press 1 to be inconvenienced, press 2 to be condescended to, and so forth. If there are many choices, what is the best way to organize them? Should you build a deep hierarchy with lots of little menus that each offer just a few options? Or is it better to flatten teh structure into a few long menus? In this situation a reasonable goal is to minimize the number of options that the wretched caller must listen to before finally reaching his or her destination. The problem is analogous to that of representing an integer in positional notation: the number of items per menu corresponds to the radix r, and the number of menus is analogous to the width w. The average number of choices to be endured is minimized when there are three items per menu.

I have no idea if it is correct. It is from the often quite interesting Group Theory in the Bedroom, and Other Mathematical Diversions, by Brian Hayes.