imc

Question

Andy the Ant lives on a coordinate plane and is currently at (-20, 20) facing east (that is, in the positive x-direction). Andy moves 1 unit and then turns 90^ left. From there, Andy moves 2 units (north) and then turns 90^ left. He then moves 3 units (west) and again turns 90^ left. Andy continues his progress, increasing his distance each time by 1 unit and always turning left. What is the location of the point at which Andy makes the 2020th left turn?

Accepted Answer

Andy makes a total of 2020 moves: 1010 horizontal (505 left and 505 right) and 1010 vertical (505 up and 505 down). The x-coordinate of Andy's final position is -20+1-3_-2+5-7_-2+9-11_-2++2017-2019_-2^1010 terms, 505 pairs=-20-2505=-1030. The y-coordinate of Andy's final position is 20+2-4_-2+6-8_-2+10-12_-2++2018-2020_-2^1010 terms, 505 pairs=20-2505=-990. Together, we have (x,y)=(-1030, -990).