Problem of the Week #54: Monday July 15th, 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 problem is inspired by this twitter post.
Starting with 3, let’s double it and add one (or 2n+1), then modulate the answer by 14 (divide by 14 and take the remainder). If we repeat these two steps, we get the first cycle.
Notice how if we start with 4, we still end up with another cycle eventually, and 4 is not part of the cycle.
a). List out all possible cycles here.
b). Notice the cycles in the previous examples are 3 numbers long. What number should we start with to reach the shortest cycle? How about if we modulate by a different number instead of 14?
c). If we modulate by another natural number n instead of 14, what proportion of the possible starting numbers between 1 and n are included in the cycles?
d). For a given natural number n, if we modulate by n, how many different cycles can we have?
e). For given natural numbers a, b, and n, if we start with a number between 1 and n, multiply by a, add b, and modulate by n at each step, what proportion of the possible starting numbers would be included in the cycles?
f). For given natural numbers a, b, and n, if we start with a number between 1 and n, multiply by a, add b, and modulate by n at each step, how many different cycles can we have?
g). For given natural numbers a, b, and n, if we start with a number between 1 and n, multiply by a, add b, and modulate by n at each step, how do we know if there is a cycle that has only one number?
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.