At a party, my wife set 24 place cards on our round table, but everybody sat down randomly, and nobody was in the right place.

My wife commented: "let's turn the table round, until at least two will be in the right place."

Can this always be so, or was it just a fluke?

Not as good as Paul's solution, but I like it.

Label the people around the table 1 to 24.  Let d(x) be the label of the person whose card is in front of person labelled x.  Then x-d(x), for x=1,...,24 produces 24 numbers, none of which is zero.  Hence, modulo 24, some two of them, say x-d(x) and y-d(y), x not equal to y, are the same number, s, mod 24.  So, when the table is shifted s places, the people labelled x and y will be in the right place (if s=13 or s=-11, shifting 13 or -11 places puts the table in the same position).

Posted by McWorter on 2005-07-27 15:40:54