Further hopping with toads and frogs (Q540055)

The game ``toads and frogs'' was invented by Richard Guy and extensively discussed by \textit{E. R. Berlekamp, J. H. Conway} and \textit{R. K. Guy} [Winning ways for your mathematical plays. Vol. 1: Games in general. Vol. 2: Games in particular. Academic Press (1982; Zbl 0485.00025)]. The game is played on a \(1\times n\) strip, with each square on the strip occupied at any time by toad (T) or frog (F), or is empty (blank) (B). Left plays \(T\) and right plays \(F\). On its move, \(T\) may move to the square immediately on its right, if that square is blank, and \(F\) to the square immediately to its left, if that is empty. If \(T\) and \(F\) are adjacent they have the option to jump over one another, in their designated directions, provided they land on an empty square. If \(X\) is \(T,\, F\) or \(B\), \(X^a\) denotes a sequence of \(a\) \(X\)s, so e.g., \(T^2B^3F^4\) represents \(TTBBBFFFF\). Starting positions are those where all \(T\) are rightmost and all \(F\) are leftmost. Some positions were analyzed in ``Winning ways''. \textit{J. Erickson} [in: Richard J. Nowakowski (ed.), Games of no chance. Math. Sci. Res. Inst. Publ. 29, 299--310 (1997; Zbl 0872.90135)] analyzed more general positions, and made six conjectures. Some of these have since been proved. The current paper includes a counterexample to Erickson's conjecture E4: \(T^aB^aF^{a-1}=1\) or \(\{1|1\}\) for all \(a\geq1\), and analyzes some further positions where there are variables on both \(T\) and \(F\), as e.g., \(T^aB^2F^b\). The paper concludes with five new conjectures.