Problem of the Week #52: Monday July 1st, 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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This problem is inspired by this twitter post.
In the following puzzle, each number indicates the distance (vertically, horizontally, or a combination of both) this tile needs to move. A solution is a rearrangement of all the tiles in the grid that satisfies all the numbers where none of the tiles overlap.
For example, here are two ways to solve the 2×2 puzzle below:
a). Given an mxn grid, how many Jumping Tiles puzzles are there that have valid solutions?
b). Given an mxn grid, what’s the biggest number of solutions a puzzle in the grid can have?
c). Given an mxn grid, how do we find the puzzle(s) with the biggest number of solutions?
d). Given an mxn grid, how many puzzles have a solution that includes all tiles in a cycle?
e). Given an mxn grid, how do we tell if a puzzle has a solution that includes all tiles in a cycle?
f). Given an mxn grid, how do we tell if a puzzle has a solution without trying out a solution?
g). Is it possible to have a 1xn puzzle with more than one solution?
h). Is it possible to have a solvable puzzle where the sum of all the numbers is odd?
i). Share your own problem inspired by this one.
j). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.