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Abstract:We consider special cases of a modified version of the Tower of Hanoi puzzle and demonstrate how to find upper bounds on the minimum number of moves that it takes to complete these cases.

Let a be a positive integer and let k be an arbitrary, fixed positive integer. We define a generalized Fibonacci-type polynomial sequence by G k,0 (x) = −a, G k,1 (x) = x − a, and G k,n (x) = x k G k,n−1 (x) + G k,n−2 (x) for n ≥ 2. Let g k,n represent the maximum real zero of G k,n. We prove that the sequence {g k,2n } is decreasing and converges to a real number β k. Moreover, we prove that the sequence {g k,2n+1 } is increasing and converges to β k as well. We conclude by proving that {β k } is decreasing and converges to a.