Chutes and Ladders: on some sequences inspired by 2017 Putnam A1

TL;DR

This paper explores a mathematical problem involving sequences of natural numbers closed under specific transformations, using graph theory to analyze their structure and element appearance order.

Contribution

It reformulates the problem as a graph theory question, generalizes the original problem, and determines the sequence of element appearances.

Findings

Characterization of the set of numbers closed under the given maps

Development of a graph-based framework for the problem

Determination of the first appearance of each number in the sequence

Abstract

The first problem of the 2017 Putnam competition was to characterize a set of natural numbers closed under both the square-root map $n^2 \mapsto n$ and the "add 5 and square" map $ n \mapsto (n+5)^2$. We reframe this as a problem on an infinite directed graph, using this framing both to generalize the problem and its solution, as well as to determine the first appearance of each number in this set under a row-wise algorithm that outputs all its elements.