Prologue

It appears that there was a club and the president decided that it would be nice to hold a dinner for all the members. In order not to give any one member prominence, the president felt that they should be seated at a round table.

But at this stage he ran into some problems. It seems that the club was not all that amicable a little group. In fact each member only had a few friends within the club and positively detested all the rest. So the president thought it necessary to make sure that each member had a friend sitting on either side of him at the dinner.

Unfortunately, try as he might, he could not come up with such an arrangement. In desperation he turned to a mathematician. Not long afterwards, the mathematician came back with the following reply.

‘It's absolutely impossible! However, if one member of the club can be persuaded not to turn up, then everyone can be seated next to a friend.’

‘Which member must I ask to stay away?’ the president queried.

‘It doesn't matter’, replied the mathematician. ‘Anyone will do.’

‘By the way, if you had fewer members in the club you wouldn't be faced with this strange combination of properties.’

So the president, on some pretext, excused himself from the dinner and was easily able to seat the members of the club so they all had a friend on either side.

How many club members were there? Who likes whom and who dislikes whom? Show that the solution is unique (to within the obvious symmetries).