Let's rotate the walk of the mad robot 45 degrees to the left. So it walks up or down along the y-axis and to the right or left along the x-axis. Each time the robot turns 90 degrees it steps forward half the previous step. After the 21e step it stops, because that step is smaller than 0.001. But to calculate easily we let the robot go on forever, after all it doesn't go forward anymore (the steps are too small) and only turns around its axis (it really is a mad robot). Upwarts along the y-axis the robot walks 1000(1 + 1/2^4 + 1/2^8 + ....). An infinite geometric series between the parenthesis. The result is 1000 x 16/15. Downwarts along the y-axis:
250(1 + 1/2^4 + 1/ 2^8 + ...) = 250 x 16/15. The total walk along the y-axis is also 750 x 16/15 = 800. Along the x-axis, the robot starts with 500, so the covered way along the x-axis is 400, half the covered way along the y-axis. Therefore, the robot reaches the point (400,800). But remember, initially we did rotate its walk, so we have to rotate back 45 degrees. The final destination of the mad robot is (600√2,200√2) or (849,283).

## The mad robot

Let's rotate the walk of the mad robot 45 degrees to the left. So it walks up or down along the y-axis and to the right or left along the x-axis. Each time the robot turns 90 degrees it steps forward half the previous step. After the 21e step it stops, because that step is smaller than 0.001. But to calculate easily we let the robot go on forever, after all it doesn't go forward anymore (the steps are too small) and only turns around its axis (it really is a mad robot). Upwarts along the y-axis the robot walks 1000(1 + 1/2^4 + 1/2^8 + ....). An infinite geometric series between the parenthesis. The result is 1000 x 16/15. Downwarts along the y-axis:

250(1 + 1/2^4 + 1/ 2^8 + ...) = 250 x 16/15. The total walk along the y-axis is also 750 x 16/15 = 800. Along the x-axis, the robot starts with 500, so the covered way along the x-axis is 400, half the covered way along the y-axis. Therefore, the robot reaches the point (400,800). But remember, initially we did rotate its walk, so we have to rotate back 45 degrees. The final destination of the mad robot is (600√2,200√2) or (849,283).

Hub Boreas