Problem of the Week #30: Monday July 31st, 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 a rectangular grid made of unit squares. We choose one square in the grid, say the one in the 3rd row and 4th column, as shown by the shaded region below:
a). How many squares in this 6×4 grid include this space in their areas?
b). How many rectangles in this 6×4 grid in clued this space in their areas?
c). If the shaded region expands to include the unit square on its right, covering a total area of a 2×1 rectangle, how many squares in this 6×4 grid include the entire shaded region in their areas?
d). How many squares in this 6×4 grid include at least a part of the shaded region from part c) in their areas?
e). How many rectangles in the grid include the entire shaded region from part c) in their areas?
f). How many rectangles in the grid include at least part of the shaded region from part c)?
g). In an mxn grid, with a pxq shaded region whose top left most unit square is in the i-th row and j-th column, where m, n, p, q, i, j are all natural numbers with values that fit this context (ie p≤m, etc), how many squares are there in the grid that include the entire shaded region in their areas?
h). How many rectangles fit the conditions in part g)?
i). How many squares include at least part of the shaded region in part g)?
j). How many rectangles include at least part of the shaded regions in part g)?
k). Share your own problem inspired by this one.
l). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.