Problem of the Week #45: Monday November 13th, 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.
This problem is inspired by this geometry puzzle on Twitter. The original problem asks to find the ratio between two equilateral triangles fitted into a square. Let’s generalize it.
a). Find the ratio of areas between the blue and the orange triangles in terms of the indicated angle if the triangles are not equilateral but isosceles, like this:
b). Find the ratio of volumes between the blue square pyramid with equilateral triangular faces and the orange square pyramid with equilateral triangular faces if they are fitted into a cube as shown:
c). Since this is a cube, the orange square pyramid has space to slide around a bit, and some of its possible positions result in an overlap of one of its equilateral triangular faces with a blue one. Find the maximum possible area of the overlap.
d). Find the ratio of volumes between the two square pyramids if the faces are isosceles instead of equilateral. Choose a variable (an angle, a length, etc) to express the ratio of volumes.
e). Find the maximum possible area of overlap in part c) if the square pyramids have isosceles triangular faces instead of equilateral.
f). Find the ratio of volumes between the pyramids if their bases and all their sides are equilateral triangles.
g). Find the maximum overlap area in the case of part f).
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.