Position three security guards in a square room as shown in the figure below (the guards are represented by X's). If each guard has responsibility for the portion of the room that is closest to him, then each guard will be responsible for a third of the room, and the distance from each guard to the point in his area furthest from his position is the same for each guard (in this case Sqrt[10]/6 times the length of one side of the room).
This month's problem is to reposition the guards so that the conditions above hold, but the distance from each guard to the point in his area furthest from him is less than in the example above.
Source: Hugo Steinhaus