Problem of the Week #40: Monday October 9th, 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.
We have explored previously a problem about tiling square grids using tetrominoes and a problem about the possible shapes of polynominoes. Now let’s consider tiling square grids using polynominoes of any number of units. For example, here is one way to tile a 4×4 square grid using pentaminoes (polynominoes with 5 units):
Notice that as with the tetrominoes problem, we need to remove one space (gray) from this grid for it to be possible to be tiled using pentaminoes.
a). Move the removed space to one of the four spaces at the center, and then find a way to tile the resulting 4×4 grid.
b). How many ways are there to tile this 4×4 grid using pentaminoes with one space removed?
c). What is the minimum number of spaces we need to add or remove to make it possible for a 7×7 square grid to be tiled using pentaminoes?
d). How many ways are there to tile this 4×4 grid using polynominoes of any sizes (not restricted to one size) without adding or removing spaces?
e). What is the minimum number of spaces we need to add or remove to make it possible for an nxn square grid to be tiled using polynominoes with k units, where n and k are natural numbers?
f). How many ways are there to tile an nxn square grid using polynominoes with k units and the minimum number of add or removed spaces, where n and k are natural numbers?
g). How many ways are there to tile an nxn square grid, where n is a natural number, using polynominoes of any sizes (not restricted to one size)?
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.