The Tower of Hanoi game is very useful in understanding the Recurrence relation.
It is a game of moving N disk between 3 needles. However, there are some rules we must follow before moving a disk from one needle to second needle.
Rule 1: Move only one disk at a time and it must be a top one.
Rule 2. Cannot move a larger disk on top of smaller one.
Rule 3: Number of moves should be minimum.
Let see how this works for 3 disk Tower of Hanoi game and using output of the game we can find solution to Tower of Hanoi game for N disk.
Initially , our disks stacked in needle 1 and other two remain empty. Each time we move a disk it is counted as 1.
Move top disk from needle 1 to needle 3.
Move top disk from needle 1 to needle 2.
Move top disk from needle 3 to needle 2.
Move top disk from needle 1 to needle 3.
Move top disk from needle 2 to needle 1.
Move top disk from needle 2 to needle 3.
Move top disk from needle 1 to needle 3.
So we took total 7 Moves to shift all 3 disks.
Let the total number of disk move be H.
If we want to compute total count for 4 disk Tower of Hanoi game.
There minimum 15 disk move required for 4 disk Tower of Hanoi game.
We can derive a much easier formula for
How we arrived at this solution, we leave it as an exercise for you.