How's your maths?

Some years ago I bought a puzzle to amuse myself whilst on a long and boring train journey. I’m not going to tell you what the puzzle is called, you can find that out for yourself if you’re that interested. I’m sure that this puzzle is like no other you have enjoyed solving. Well anyway, it’s a wooden puzzle consisting of 8 discs on a board with 3 pegs, just like the illustration left.
You have to move the discs from the middle peg (although you can start on any peg you wish) to 1 of the outer pegs and back again. You have to try and complete the puzzle in so many moves, although you can take longer if you wish, and there are only two rules to this puzzle, which are.
1) You may only move one disc at a time.
2) You cannot place a larger disc on a smaller one.
When you start with the 8 discs, then the minimum number of moves that you could complete this puzzle in is 255.

Now, if you want some more fun, then there is a formula that you can use to calculate how many moves it would take to complete the puzzle if it had any number of discs. Ok...Let us assume that the number of discs is represented by the variable n. So, in the following formula, T corresponds to the number of moves required in relation to the number of discs used in the puzzle. Here is the formula. T(n)=2n-1(2 to the power of n).
OK, for all you mathematicians’ out there – here’s your test, why not try and work out the number of moves it would take if you had 9 discs instead of 8.
5 discs = 31   moves 2power5-1
6   "     = 63   moves 2power6-1
7   "     = 127 moves 2power7-1
8   "     = 255 moves 2power8-1
9   "     = ?
I have to tell you that if you haven’t played this before, then you'll find it both absorbing and fun. What is even more astonishing is the maths involved in working out how long it would take you to complete the puzzle if you had a total of 64 discs, as suggested by the French mathematician who invented the game - well, he did indeed work it out and the figures involved are truly astronomical.
I went to my local village hall where members of the local raffle and chess club were competing with each other to be the first to solve it, and I told everyone of my sucess with the puzzle...I was the only one to complete it in the minimum time...so excited was everyone that two members hoisted me up and tried to carry me aloft in celebration around the hall, but sadly they both colapsed under my weight and were taken to A and E shortly afterwards and released - heavily bandaged - the next day. Still, we all had a good laugh about it.

Here's a clue to how long it would take if you played with 64 discs...how old is the Universe...and that's just a start!

Just mouse over this pic