Question 18937
first let me show how to find the answer for the next case (6 rings)if we know the answer for the previous case (5 rings)
answer for 5 rings =31...ok ...now let us find the answer for 6 rings .
to start with assume the additional new ring (6th.in this case and the biggest) is fixed to the bottom.so we have to move only 5 smaller rings. let us say we move t to the intermediate stand(it should not matter to which stand we move them  as long as we use 3 stands only)which requires 31 moves.now having finished this let us remove the new sixth ring which is freed now and put it on the required final stand.this requires only 1 move.now let us imagine it as fixed and being biggest it can remain there at the bottom without any problem.
now let us transfer the 5 smaller rings to this stand which requires 31 moves again.so total number of moves required to move 6 rigs=31 +1 +31=63 moves..if we put it in a formula with the notation that we need M(5) moves to move 5 rings and M(6)moves to move 6 rings... the formula we got is 
M(6)=M(5)+1+M(5)=2M(5)+1=2*31+1=63....you can check the correctness of this 
farmula with the other answers you got .