Problem of the Week #27: Monday July 10th, 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.
Consider the widely known problem of counting the number of squares in a square grid. Today we will try to extend the problem.
a). How many squares are there in the following 4×4 square grid?
b). How many squares are there in an nxn square grid where n is a natural number?
c). How many squares are there in an mxn grid where m and n are natural numbers?
d). Find the natural number n such that there are exactly 55 squares in the nxn grid.
e). Find a pair of natural numbers m and n such that there are exactly 50 squares 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 squares 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 squares 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.