Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire

TL;DR

This paper explores a novel integer procedure called the strange root, its connection to Tchoukaillon solitaire, and analyzes conditions under which certain values are strange roots, proposing related conjectures.

Contribution

It introduces the concept of strange roots, links it to Tchoukaillon solitaire, and analyzes the inverse problem of identifying integers with a given strange root.

Findings

Strange roots are linked to Tchoukaillon solitaire configurations.

The paper characterizes when a value is the strange root of up to two integers.

A conjecture relating strange roots to game states is proposed.

Abstract

In this paper we examine a procedure that, on starting with an integer $n$, results in a pair of equal integers that are no greater than $n$. We call the resulting value the \textit{strange root} of $n$ and we show how this strange-root-finding procedure is intimately linked to the game of Tchoukaillon solitaire. We analyze the strange-root-finding procedure in reverse to determine when a prescribed value is the strange root of at most two integers. We present a conjecture about strange roots and translate this conjecture into one involving Tchoukaillon solitaire.