Problem of the Week #64: Monday September 23rd, 2024
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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a). In this 4×4 grid, if we choose two squares, for which pairs of squares can we find a path from one square to the other, moving only vertically or horizontally at each step, and visiting every square in the grid exactly once?
b). In a given rectangular grid of any dimensions, how can we tell if a given pair of squares have such a path from one to the other that visits every square in the grid exactly once? Can you prove it?
c). If the grid is not rectangular, how can we tell if a given pair of squares have such a path through the grid?
d). How can we tell from the shape of a grid if it has a loop (path where two end points are neighbours and visits each square exactly once)?
e). How can we tell from the shape of a grid if it has a trail (path where two end points are not neighbours and visits each square exactly once)?
f). How can we tell from the shape of a 3 dimensional grid if it has a loop or a trail?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.