Problem of the Week #47: Monday November 27th, 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 problem on twitter.
The original problem asks to cut a square by drawing straight lines between any number of midpoints of sides and corners of the square such that a resulting region is 1/5 of the area of the whole square.
Like this:
It was intriguing because why 1/5? Where did the 5 come from? So I did some digging and came up with some more questions.
a). Find one solution to the original problem.
b). How many ways are there to get an area of 1/5 of the square as specified in the original problem?
c). How many different unit-fractional areas of the square can we get using the given points?
d). How many different unit-fractional areas of the square can we get by dividing each side into equal thirds instead? Like this:
e). Can we get a region that is 1/7 of the area of the square without dividing each side into 7 equal parts?
f). For a given natural number n, find the smallest natural number k such that we can obtain a region with area 1/n of the square by dividing each side of the square into k equal parts.
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.