Problem of the Week #55: Monday July 22nd, 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 draw a smaller square inside a unit square by dividing out a fixed fraction of each side and connecting the dividing point to an opposite angle. For example, this is what the smaller square would look like if we divide each side of the large square in half:
a). What’s the area of the smaller square?
b). Find a formula for the area of the smaller square if a is any fraction between 0 and 1.
c). Does the formula you developed from the last question still work for a negative a?
d). Find the formula for the area of the smaller pentagram or hexagram if a is any fraction between 0 and 1. Does this formula still work for a negative a?
e). If we pick a face of the cube (top in this case), connect two adjacent midpoints of the sides, and then connect them to the closest vertex on the opposite face, we get a triangular cut like the shaded part. Notice the cut in the background that’s made from the opposite face is parallel to the one we made. There are 24 cuts on the cube that are congruent (exactly the same shape) as this, 12 pairs of parallel cuts. Is any of them perpendicular to the cut we made?
f). Is there a set of cuts on a unit cube that would result in a smaller cube that doesn’t have any face parallel to the original outer cube?
g). If a solution to part f) exists the formula for the volume of the inner cube.
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.