I start my pen at the origin of the plane. On my first turn I draw a line of length 1, and the second I draw a line of length 2, and on the nth turn, I draw a line of length n. The line on turn n+1 starts at the endpoint of the line on turn n. Lines can intersect and overlap each other. Is it possible to draw an NxN square for some integer N?
I'm not quite sure how to start attacking this problem. Any hints or guidance would be nice.
Use the fact that stepping $n$ units in one direction and $n+1$ in the opposite direction results in a net change of one unit, so as long as the steps don't get out of bounds, one can get to the right places one step at a time. Here's one solution for a $78\times 78$ square, where positive values indicate steps in the positive direction of the $x$ or $y$ axis and negative values indicate steps in the negative direction, starting at $(0,0)$:
Along the $x$-axis:
$[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
Along the vertical line through $(78,0)$:
$[13, 14, 15, 16, 17, -18, 19, -20, 21, -22, 23]$
Along the horizontal line through $(78,78)$:
$[-24, -25, -26, 27, -28, 29, -30, 31, -32]$
Along the $y$-axis:
$[-33, -34, 35, -36, 37, -38, 39, -40, 41, -42, 43, -44, 45, -46, 47, -48, 49, -50, 51, -52, 53, -54, 55, -56]$