Problem of the Week #84: Monday March 31st, 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 was inspired by Round 2 of the 1972 British Math Olympiad, Q1
Let’s arrange all 3 digit binary codes in a list starting with 000 such that each code differs from its previous code by only 1 digit.
a). Find one such arrangement of 3 digit binary codes.
b). If we arrange all 3 digit binary codes in a list, starting with 000, such that each code differs from its previous code by only 1 digit, is it possible for 101 to be the last code on the list?
c). How many ways are there to arrange all 3 digit binary codes in a loop, starting with 000, such that each code differs from its previous code by only 1 digit (including the first and last codes)?
d). Prove that we can always arrange all n digit binary codes in a list starting with all 0s such that each code differs from its previous code by 1 digit.
e). Find all possible last codes if we arrange all n digit binary codes in a list starting with all 0s such that each code differs from its previous code by 1 digit.
f). How many ways are there to arrange all n digit binary codes in a loop, starting with all 0s, such that each code differs from its previous code by only 1 digit (including the first and last codes)?
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.