Once, a number of couples, each one of them happening to be composed of a mathematician and an athlete (hence 'mathletic'), wanted, in order to diversify communication, to sit down at opposite sides of a (rectangular) dinner table randomly, in such a way that mathematicians face athletes. A certain athlete proposed: let's simply not allow any couple members to face one another, this increases average distance between couple members by 0.35 chairs, and such will diversify communication even more.
How many couples were there?
Assume no mathematician is also an athlete, and vice versa, or, at least, any couple member is always strictly more mathematician than athlete, or, strictly more athlete than mathematician (so one member can be labeled 'm' and the other one 'a', or vice versa).
The proposal got accepted unanimously. :-)
'distance' between 'couple members' is measured in number of chairs away to the left or to the right. Below some examples of distance 2 between couple members 'm' and 'a' of couple '(m,a)' from 'm' point of view:
2|1|0|1|a 1|0|1|2|3 table or table or ... 2|1|m|1|2 1|m|1|a|3
This question, which I once came up with, was recently originally posted on 'mathematics' website, but, I was kindly suggested to rather post it here on the 'puzzle' website.