## 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?

### Comments

http://imgur.com/xYt9uRj I worked this problem using a basic fundamental math. I got the solution of 200 squares that the sailor can travel on.

You must have counted wrong, because the maximum can be no more than 196 by the following argument:

Consider the parity of the coordinates of the squares in the path. Taking every second square, there is alternation between EE/OO and EO/OE, where E and O stand for even and odd. If there are 15 rows, only 7 have even parity, and similarly with columns, so there are only 49 squares with parity EE, and hence a maximal payh length of 196.

@Purin is wrong. You missed two parts of the question. 1) The sailor can't enter and exit a square in a straight line. You did that going from 37 to 38. 2) The sailor must end on the square that he started one. You have him starting in a corner and ending in the middle.

The highest I found in 176. http://nyccami.org/what-do-you-do-with-a-dizzy-sailor/

I got 172. A question I had as an extension was to figure out this problem for each size grid from 2x2 - 15x15. And...is there a pattern (or function) in the largest number of squares for each size grid. Solange