2019 OIM Problems/Problem 5

Problem

Don Miguel places a game piece on one of the $(n + 1)^2$ vertices defined by a $n \times n$ board. A move consists of moving the piece from the vertex where it is located at an adjacent vertex in one of the eight possible directions: $\uparrow, \downarrow, \to, \gets, \nearrow, \searrow, \nwarrow, \swarrow,$ as long as it does not go off the board. A tour is a sequence of moves such that the game pieces has been at each of the $(n + 1)^2$ vertices exactly only once. What is the greatest number of diagonal moves ($\nearrow, \searrow, \nwarrow, \swarrow,$) that you can have in a tour?

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions