Problem of the Week #58: Monday August 12th, 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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Here’s one way to put a pair of 1s, a pair of 2s, and a pair of 3s in a row such that the pair of 1s have one number between them, the pair of 2s have two numbers between them, and the pair of 3s have three numbers between them:
a). Arrange pairs of natural numbers 1 to 4 in this way.
b). Is this doable for pairs of natural numbers 1 to 5? What about 1 to 6? 1 to 7?
c). For which natural numbers n is this doable for pairs of natural numbers 1 to n? Why?
d). For pairs of natural numbers 1 to 5, if we remove one pair of numbers so that it’s possible to arrange the rest in this way, which pair can it be?
e). What’s the maximum number of pairs of numbers we can remove from pairs of natural numbers 1 to 6 so that the remaining numbers can still be arranged in this way?
f). Given any set of distinct pairs of identical natural numbers, how can we tell if it can be arranged in this way?
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.