### Assignment #5

**Due: 2 December, 1997**

**1.** A level surface is illuminated by a single lamp. The intensity of illumination of a small piece of the surface is directly proportional to the cosine of the angle of incidence of the light and is inversely proportional to the square of the distance from the light source. If we draw a circle 10 *m* wide on the surface, how high above the centre of the circle must we place the lamp to maximize the intensity of illumination at the edge of the circle? *[8]*

**2.** A two-player game (in which the players take turns making moves) is considered to be *finite* if it cannot go on forever when played by the rules. For example, tic-tac-toe is finite. So is chess, thanks to some obscure rules about about the game being an automatic draw if no one captures a piece, moves a pawn, or delivers checkmate in a certain number of moves. (50 by each player in most situations, but there are a few exceptions.) The two-player game SUPERGAME is played as follows: the first player chooses a finite two-player game, which the two players proceed to play out with the second player going first. Is SUPERGAME itself finite? Why or why not? *[2]*

Solution to Assignment #5

Department of Mathematics
Trent University

Maintained by Stefan Bilaniuk. Last updated 1998.08.22.