Consider the following impartial combinatorial game. Two boxes each contain a nonzero number of chips. Two player alternate turns. A turn consists of a player choosing a box and emptying it, removing 0 or 1 chip from pile in the other box, then dividing the remaining chips so that both boxes again have a nonzero number of chips. Thus, (1,1) is the only terminal position. Required:
Show that (x, y) is a P-position if and only if x mod 3 = 1 and y mod 3 = 1.

Drag the tiles to the correct boxes to complete the pairs. match the exponential functions with their horizontal asymptotes and y-intercepts. f(x) = 7x − 4 f(x) = 3x+2 + 4 f(x) = 9x+1 − 4 f(x) = 2x + 4 horizontal asymptote and y-intercept exponential function horizontal asymptote: y = 4 y-intercept: (0, 5) arrowboth horizontal asymptote: y = -4 y-intercept: (0, 5) arrowboth horizontal asymptote: y = -4 y-intercept: (0, -3) arrowboth horizontal asymptote: y = 4 y-intercept: (0, 13) arrowboth