2006 OIM Problems/Problem 3

Problem

The numbers $1, 2, 3,\cdots , n^2$ are placed in the squares of an $n \times n$ grid, in some order, one number per box. A game piece is initially located on the square with the number $n^2$ . At each step, the game piece can advance to any of the squares that share a side with the square where it is located. First, the game piece travels to the square with the number 1, and to it takes one of the shortest paths (with fewer steps) between the square with number $n^2$ and the square with number 1. From the square with number 1 travel to the square with number 2, from there to the square with the number 3, and so on, until you return to the initial square, taking the shortest route on each trip. The complete route takes the token $N$ steps. Find the smallest and largest possible value of $N$ .

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

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2006 OIM Problems/Problem 3

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