smc

Question

The taxicab distance between points (x_1, y_1) and (x_2, y_2) in the coordinate plane is given by |x_1 - x_2| + |y_1 - y_2|. For how many points P with integer coordinates is the taxicab distance between P and the origin less than or equal to 20?

Accepted Answer

Let us consider the number of points for a certain x-coordinate. For any x, the viable points are in the range [-20 + |x|, 20 - |x|]. This means that our total sum is equal to align* 1 + 3 + 5 + + 41 + 39 + 37 + + 1 &= (1 + 3 + 5 + + 39) + (1 + 3 + 5 + + 41) & = 20^2 + 21^2 & = 29^2 &= , 841. align*