  We study two impartial games introduced by Anderson and Harary. Both games
are played by two players who alternately select previously-unselected elements
of a finite group. The first player who builds a generating set from the
jointly-selected elements wins the first game. The first player who cannot
select an element without building a generating set loses the second game. We
determine the nim-numbers, and therefore the outcomes, of these games for
symmetric and alternating groups.
