More dipping into really. It's Mathematical Puzzles: Revised Edition, by Peter Winkler. It's rather good, but I only stumbled across it by accident. Somehow (I can't remember how) I read a Guardian newspaper puzzle column and it contained this gem from Winler's book:

There are six distances among four points and in general they are all different, as seen the left-hand sketch below. The sketch on the right however shows a configuration in which only two different distance are represented; the side of the square are one length, and the diagonals the other. This is one solution to our puzzle then, but it is not the only one. How many others can you find? In Winkler's words "There are more of them that you probably think".


If you want to see the full solution the Guardian column provides a link to it.

This got me thinking about related problems.  With five points you can still keep to two distances but there is only one way (I think.) What about six points? Well, three different distances is the minimum possible number, but how many ways can it be done in? So far I have found four but I haven't tried yet to prove these are the only ones. Here are four I've found, with descriptions.