Problem of the Week #49: Monday December 11th, 2023 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. Please register for an account if you would like to join the discussion below or share your own problems.
This problem is inspired by this twitter post.
The original problem asks for a combination of 1×2 and 1×3 rods that can tile a 5×5 square. Tiling problems have a tendency to get complicated quickly, so let’s start small.
a). How many ways are there to tile a 3×3 square using 1×2 or 1×3 rods?
b). How many ways are there to tile a 4×4 square using only 1×2 rods?
c). How many ways are there to tile a 4×4 square using 1×2 and 1×3 rods (at least one of each)?
d). How many ways are there to tile a 4×4 square using 1×2, 1×3, and 1×4 rods (at least one of each)?
e). How many ways are there to tile a 4×4 square using rods of any sizes (1xn)?
f). How many combinations of different sized rods are there for tiling the 5×5 square? For example, the figure for part d) uses this combination of rods: 1 of 1×4, 2 of 1×3, and 3 of 1×4.
g). How many ways are there to tile a 5×5 square using rods of any sizes?
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.