40. There is a moving-knife version of the Banach- Knaster procedure that is due to Dubins and Spanier. To describe it, we picture the cake as being rectangular and the procedure beginning with a referee holding a knife along the left edge, as illustrated below.

Assume, for the sake of illustration, that there are four players (Bob, Carol, Ted, and Alice). The referee starts moving the knife from left to right over the cake (keeping it parallel to the position in which it started) until one of the players (assume it is Bob) calls “cut.” At this time, a cut is made, the piece to the left of the knife is given to Bob, and he exits the game. The knife starts moving again, and the process continues. The strategies are for each player to call “cut” whenever it would yield him or her a piece of size at least one-fourth.