However, I was wondering whether anyone could provide any clues on how you go about working out the minimum number of moves for... say, 100 disks without knowing the previous answer. I can't even seem to get started on finding the formula...

Thanks very much.

I do have a story about it, though. In 1884, a certain man called de Parville described in La nature, part I, pp 285f, what must be the ultimate Tower of Hanoi.

Paraphrasing, the dome of the great temple at Benares marks the centre of the world, and beneath it a huge Tower of Hanoi affair with three diamond needles each a cubit high, and 64 discs of gold. Day and night, the priests move discs from needle to needle, and when the "game" is complete, the world will come to an end.

If the priests work in shifts on a 24-hour schedule, transfer one disc per second and never, ever make a mistake, that works out at almost six billion centuries, making this the most optimistic prophecy of the end of the world on record. When you find the formula, you can check.

For reference: the usual scientific guess for the end of the world -- when our sun expands to a red giant and swallows up the earth -- is about eight billion years.

https://www.cut-the-knot.org/recurrence/hanoi.shtml

if you want to figure it out for yourself and just want a clue, a good aproach in solving recurences is to carefully "watch yourself over the shoulder" while figuring. When you have number a, what do you do with it to get a and in which "prominent" sequences is the relationship between an element and its successor roughly similar?

good luck

2^^n - 1 (2 to the n (n rings) then subtract 1 from that answer)

eg, 3 rings should take minimum number of moves:

2^^3 = 8
8 - 1 = 7

7 moves for three rings

2^^100 = 1.3 x E30
then subtract 1 (which really doesn't make a difference.)

Here's something to think about. If it takes one second to make a move, and you have 27 rings, you don't have enough time in your life to complete the process, even if you make the minimum number of moves.