Problem of the Week #39: Tuesday October 3rd, 2023 Sorry for the delay this week. Family emergency. 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.
If we separate all tetrominoes according to their perimeters, we would have two groups:
Tetrominoes with perimeter 10:
Tetromino with perimeter 8:
For the purpose of this entire problem any rotation and reflection of a polynomino is considered the same polynomino. If a polynomino contains a hole, then the perimeter of the hole is considered part of the perimeter of the polynomino.
a). How many possible perimeters are there for a pentamino (polynomino with 5 units)?
b). How many pentaminoes have the largest possible perimeter?
c). How many pentaminoes have the smallest possible perimeter?
d). How many units is required for a polynomino to contain a hole?
e). What is the perimeter of the smallest polynomino to contain a hole?
f). How many possible perimeters are there for a polynomino with n units, where n is a natural number?
g). How many polynominoes with n units, where n is a natural number, have the largest possible perimeter?
h). How many polynominoes with n units, where n is a natural number, have the smallest possible perimeter?
i). What is the maximum number of holes a polynomino with n units can contain, where n is a natural number?
j). Share your own problem inspired by this one.
k). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.