Problem of the Week #95: Monday September 15th, 2025
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 problem is inspired by the 1975 Canadian Math Olympiad Q6i.
a). 4 people (ABCD) are at a table for a meeting. Sketch a way to place the name cards (abcd) such that they’re all mismatched. Then rotate the table such that at least 2 people have the right name cards. Is this always possible?
b). With 5 people at the meeting, sketch a way to place the name cards such that they’re all mismatched. Can you always rotate the table such that at least 2 people have the right name cards?
c). For which numbers of people is it always possible to rotate the table such that at least 2 people have the right name cards if they all start out mismatched?
d). How many different arrangements of mismatching all name cards are there for 4 people around the table? (rotating the table counts as the same arrangement)
e). How many different arrangements of mismatching all name cards are there for 5 people around the table? (rotating the table counts as the same arrangement)
f). How many different arrangements of mismatching all name cards are there for n people around the table? (rotating the table counts as the same arrangement)
g). Check out this YouTube video on the problem:
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.