Problem of the Week #41: Monday October 16th, 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.
Let’s define polycubes as three dimensional shapes made up of unit cubes connected by shared faces. For example, these are all the possible 1-, 2-, and 3-unit polycubes:
Notice that all of these polycubes are flat, meaning they can be repositioned so that they occupy only one layer.
a). Find a tetracube (4-unit polycube) that is not flat.
b). Find the pair of tetracubes that are mirror image of each other but cannot be repositioned to look exactly the same as the other. (These are called a chiral pair of tetracubes.)
c). How many different tetracubes are there in total?
d). How many flat pentacubes (5-unit polycubes) are there?
e). How many chiral pairs of pentacubes are there?
f). How many different pentacubes are there in total?
g). How m any chiral pairs of n-unit polycubes are there, where n is a natural number?
h). How many different n-unit polycubes are there in total, where n is a natural number?
i). Share your own problem inspired by this one.
j). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.