Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire

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 $\mathit{\text{strange}}\mathit{\text{root}}$ 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.