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What a lame rook

Source: 44th International Tournament of Towns, Senior O-Level P5, Fall 2022

February 16, 2023
combinatoricsdominoesboardTournament of Towns

Problem Statement

A 2N×2N2N\times2N board is covered by non-overlapping dominos of 1×21\times2 size. A lame rook (which can only move one cell at a time, horizontally or vertically) has visited each cell once on its route across the board. Call a move by the rook longitudinal if it is a move from one cell of a domino to another cell of the same domino. What is:
[*]the maximum possible number of longitudinal moves? [*]the minimum possible number of longitudinal moves?