Problem of the Week #67: Monday October 14th, 2024
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.
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This problem is inspired by this twitter post.
Let’s remove the central portion of a regular tetrahedron such that what remains are 4 tetrahedrons each with half of the side length of the original.
a). What shape is the portion that’s removed?
b). What are the volume, surface area, and total edge length of the removed portion compared to the original tetrahedron?
c). If we remove this portion from the 4 smaller tetrahedrons as well and continue the process indefinitely, what are the limits of the total volume, surface area, and total edge length of the removed portions?
d). What are the limits of the total volume, surface area, and total edge length of all the smallest tetrahedrons combined?
e). If we cut a tetrahedron from each corner of a cube as shown, what fraction of the volume is left?
f). If we cut a tetrahedron from each corner of a cube using the whole length, allowing the tetrahedrons to overlap, what shape and fraction of the volume of the cube is left?
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.