325 Thackeray Hall
Abstract or Additional Information
A monovariant is a quantity which is either non-increasing or non-decreasing, such as the number of primes up to $x$, but not the number of factors of $n$. Many challenging problems can be solved by associating the right monovariant to it; unfortunately it is often challenging to find the right quantity to study. After describing classic problems such as the Zombie Apocalypse and Conway's Soldiers we turn to recent applications. Zeckendorf proved every positive integer can be written as a sum of non-adjacent Fibonacci numbers (1, 2, 3, 5, 8, ...); using mono-variants we can show no decomposition as a sum of Fibonacci numbers has fewer summands, and discuss generalizations to other sequences. These are key ingredients in analyzing a game involving Fibonacci numbers, where we can prove Player 2 has a winning strategy but it is not known what it is. This work is joint with many students; there is a $500 reward for a constructive proof of Player 2's winning strategy!