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On ROOT INVARIANTS of PERIODIC CLASSES 1 Introduction and Ams  Form

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Can you cover this broken chessboard with 2 by 1 dominos? Can you add or subtract the numbers from 1 through 6 to get 0? Can you swap these numbers into order using an odd number of swaps? These problems all look different from first glance, but they have one major thing in common: They can be solved with invariants. So, what is an invariant? Given a process, an invariant is something that does not change through every step of it. It’s easiest to explain what that actually means with an example. Suppose we start with the numbers 1 through 6 on a chalkboard, and then begin a process such that every minute, we erase two numbers and replace them with their product, repeatedly, until there’s only one number left. Then an example of an invariant in this process would be the total product of the numbers on the chalkboard; throughout the entire process, after each minute, the total product stays the same. So when the process ends with just one number left, it’s clear that it must be the

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