The Hanoi coins game
Suppose you are given 4 pairs of coins of different diameters and you have in front of you drawn 3 squares \(S_1\) , \(S_2\) , \(S_3\) on a piece of paper anyway.
In the first two squares , \(S_1\) and \(S_2\) , you construct 2 towers of 4 coins where each coin is placed on a coin with a bigger diameter.
All coins in tower A are up , all coins in tower B are down. You play a game such that starting from this you must invert it , all coins in A should be down and all coins in B should be up by a sequence of moves which has anyway the following rules.
Anyway , consider a move to respect the following 2 characteristics:
- At every moment of a "move" you take the above coin of one of the towers and move it in one of the 2 other squares available anyway.
- Of course , you are not allowed to move a coin over another coin which has a smaller diameter anyway.
What is the minimum number of moves necessary for you to swap the 2 towers of coins ?
Insert your number as that number of minimum moves necessary.
And , for having a bonus (which almost necessary imposes itself) can you find a general understanding of what is the smallest number of moves necessary to made for any equal number of coins placed in the two towers at the beginning of the game anyway ?