Problem of the Week #61: Monday September 2nd, 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 week’s problem is inspired by last week’s problem: Nonlinear Dots
Here’s an arrangement of the numbers 1 to 4, and what it looks like plotted into a 4×4 grid such that each row corresponds to a number in the arrangement. Notice that the 3 dots on the bottom right form a straight line.
a). How many arrangements of the numbers 1 to 4 do not have 3 dots that form a line when plotted this way?
b). How many arrangements of the numbers from 1 to 5 do not have 3 dots that form a line when plotted in a grid?
c). How many arrangements of the numbers from 1 to n do not have 3 dots that form a line when plotted in a grid?
d). How many pairs of arrangements of the numbers from 1 to 4 can be plotted into the same grid such that there are no overlapping dots and no 3 dots forming a straight line?
e). How many pairs of arrangements of the numbers from 1 to 5 can be plotted into the same grid such that there are no overlapping dots and no 3 dots forming a straight line?
f). How many pairs of arrangements of the numbers from 1 to n can be plotted into the same grid such that there are no overlapping dots and no 3 dots forming a straight line?
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.