## Rubik’s Cube Game

My son Sergei invented the following game a couple of years ago. Two people, Alice and Bob, agree on a number, say, four. Alice takes a clean Rubik’s cube and secretly makes four moves. Bob gets the resulting cube and has to rotate it to the initial state in not more than four moves. Bob doesn’t need to retrace Alice’s moves. He just needs to find a short path back, preferably the shortest one. If he is successful, he gets a point and then it is Alice’s turn.

If they are experienced at solving Rubik’s cube, they can increase the difficulty and play this game with five or six moves.

By the way, how many moves do you need to solve any position on a Rubik’s cube if you know the optimal way? The cube is so complicated that people can’t always know the optimal way. They think that God can, so they called the diameter of the set of all possible Rubik’s cube positions, God’s Number. It was recently proven that God’s Number is 20. If Alice and Bob can increase the difficulty level to 20, that would mean that they can find the shortest path back to the initial state from any position of the cube, or, in short, that they would master God’s algorithm.

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1. #### Collin:

Finding an optimal set of Rubik’s Cube moves can be done on a computer with the so-called “A* Heuristic”. I’ve seen programs that do this online.

The A* Heuristic can be applied to almost every Rubik-type puzzle. Besides puzzles, it can also be applied to the plain old Euclidean Metric, which is probably the way online maps find routes.

2. #### Ken Vitoff:

I have tried the Cube Game. I did up to 6 or 7 moves, I can’t remember. But the thought process was amazing: either you got the exact same moves as was given or you went about it another way losing if going over the original moves or winning with less.

And to read that God’s number went down to 20? Absolutely fantastic!! Rubik himself would be amazed.

As a side comment, I really enjoy your website, blog and number gossip.

3. #### Vishal:

I think 20 is the upper bound on the God’s Number. It’s not THE God’s Number.