Problem of the Week #33: Monday August 21st, 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.
The cube solid below is made up of 4 unit cubes, three on the lowest level and one on the second level:
If we were to look at this solid from the front, (right) side, and top, we would see the following views:
For all questions in this problem, let’s ignore the effects of gravity, which means higher unit cubes can float in space without lower unit cubes to support them.
a). Considering all cube solids made up of unit cubes and are at most 2 units in length, width, and height, is this the only cube solid that gives this combination of front, side, and top views?
b). Find a combination of front, side, and top views for a cube solid at most 2 units in length, width, and height that fits no other cube solid.
c). How many combinations of 2×2 front, side, and top views fit exactly one cube solid?
d). How many combinations of 2×2 front, side, and top views do not fit any cube solid at all?
e). How many combinations of 2×2 front, side, and top views fit more than one cube solid?
f). What is the maximum number of different cube solids a combination of 2×2 front, side, and top views can fit?
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.