An old chestnut
goes like this:
Fourteen friends go to a restaurant together, but the restaurant cannot
seat all of them at one table so they split into two groups of six and eight.
The next time they go there, they split up into six and eight again, but
they split differently so that different friends get a chance to eat
together (at the same table, that is).
How many times must they do this before it is possible for every pair
of friends to have eaten at the same table at least once?
What about the same question for different sizes of splits? Different
numbers of friends? Ignore the degenerate cases of 0-seat tables and
the unsolvable case of two friends split between two 1-seat tables.