Problem of the Week #37: Monday September 18th, 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.
These are all the possible tetrominoes:
Let’s consider tiling square grids with these. Pieces can be rotated, reflected, and repeated as needed. For example, here are a couple of ways to tile the 3×3 grid using tetrominoes:
Notice that in the 3×3 grid there will always be one space left after all others are tiled. We can place this space anywhere on in the grid.
a). Find a way to tile the 3×3 grid if the leftover space is in the middle column of the top row.
b). How many ways are there to tile the 3×3 grid using tetrominoes and one left over space?
c). How many left over spaces do we need when tiling a 4×4 grid using tetrominoes?
d). How many ways are there to tile a 4×4 grid using tetrominoes?
e). How many left over spaces do we need when tiling an nxn grid?
f). How many ways are there to tile an nxn grid using tetrominoes?
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.