Problem of the Week #38: Monday September 25th, 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.
Polynominoes are figures made up of square units connected by edges. For example, there is only one of each of monominoes (polynominoes with 1 unit) and dominoes (2 units) and only two trinominoes (3 units), as shown below:
There are 5 tetrominoes (4 units), and 2 of them have a mirror image that cannot be rotated to overlap with its original. Let’s call these two figures sided tetrominoes.
a). Find the number of lines of symmetry for each of the tetroninoes.
b. Which of the tetrominoes have a point of symmetry?.
c). In how many ways can each of the tetrominoes be placed in a square grid? For example the first tetromino above can be place vertically or horizontally, while the last has only one orientation in the grid.
d). How many pentominoes (5 units) are there?
e). How many sided pentominoes are there?
f). How many pentominoes have exactly 4 orientations in the square grid?
g). How many pentominoes have exactly 8 orientations in the square grid?
h). How many n unit polynominoes are there, where n is a natural number?
i). How many sided n unit polynominoes are there, where n is a natural number?
j). How many n unit polynominoes have exactly 4 orientations in the square grid?
k). Share your own problem inspired by this one.
l). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.