Hanoi wants more Towers

Probability Level 4

The Tower of Hanoi is a traditional game where there are 3 pegs and nn discs arranged as seen in picture. All discs are to be moved from 1st1^{st} peg P1P_1 to 3rd3^{rd} peg P3P_3 such that only one disc is moved at once. But no disc should be kept on one smaller than it during the process. So the minimum number of moves required to transfer all discs to P3P_3 (say ana_n) is counted by an=2an1+1a_n=2a_{n-1}+1 and a1=1a_1=1 i.e. after solving becomes an=2n1.a_n=2^n-1.

If we have a Tower of Hanoi with 42 pegs, and we have an additional rule that no disc can be moved more than twice, what is the maximum number of discs such that all discs can be moved to pegs other than the first peg P1?P_1?

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