The Tower of Manaus (ToM) is a puzzle that merges the rules of the Tower of Hanoi (ToH) with the constraints of the Tower of London (ToL). It involves n distinct discs distributed across three pegs with capacities n, \(n-1\) , and \(n-2\) , respectively. Discs must be moved one at a time, maintaining size order and obeying ToH rules. The goal is to transfer all discs to the largest peg. We define a graph \(M_n\) , where vertices represent valid state and edges represent legal moves. We characterize \(M_n\) recursively based on the structure of the ToH graph \(H_n\) , showing that \(M_n\) splits into two components and can be partitioned into subgraphs T, L, and R, each related to subgraphs of \(H_{n-1}\) and \(M_{n-1}\) . We also determine the number of moves required to solve the puzzle in the worst case. Finally, we classify the vertices into good and bad, showing that \(M_n\) has \(3^n - 2(n+1)\) vertices, with precise counts for each class.
