Spring 2017, problem 34
A dizzy sailor is standing on a $15\times 15$ square tiled board. From their initial square they are able to move to any square sharing a common side. Due to the the sailor's dizziness, after every move they immediately make a left or right turn before repeating this process (that is, they are never able to enter and exit a square in a straight line). What is the largest number of squares the dizzy sailor can walk on if they are not allowed to repeat squares and the last step of their path must end at the square they started at?