Problem of the Week #28: Monday July 17th, 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.
Continuing from last week’s problem about counting squares, let’s look at the related problem of counting the number of rectangles in a grid, with extensions:
a). How many rectangles are there in the following 4×4 square grid?
b). How many rectangles are there in an nxn square grid where n is a natural number?
c). How many rectangles are there in an mxn grid where m and n are natural numbers?
d). Find the natural number n such that there are exactly 225 rectangles in the nxn grid.
e). Find a pair of natural numbers m and n such that there are exactly 90 rectangles in the mxn grid.
f). Given any natural number k, how do we determine if there is a pair of natural numbers m and n such that there are exactly k rectangles in the mxn grid?
g). Is there a natural number k for which there are more than one pair of natural numbers m and n, where m≤n, such that there are exactly k rectangles in the mxn grid?
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.