# What is the equation to solve the puzzle box Super-CUBI 324 step

The Super-CUBI puzzle box takes 324 steps to open or close it, instructions. And uses a Trinary System. It must have a solution as it would be hard to remember all the steps to take to open it. But what is it ?

• I'm very, very jealous if you have this thing. – Gracelyn Rioux May 17 '15 at 14:41
• I was very lucky to get the super cubi but the king cubi is the one I would like to have, 1536 steps Quaternary system used. – Tom May 17 '15 at 17:26

As you have said, the Super Cubi requires 324 moves to be opened. The correct sequence is listed here: In each step, you have to move one of the six faces. The numbers in the table represent the centimeters of translations. For example, 000022 means that faces #1 and #2 were moved by 2cm.

But how do you remember that sequence?
As you can see, in the first vertical half of the table the face #6 is always 0, while in the second half is always 1 (except for the last move). The face #5 is 0 for the first and last column, 1 for the second and fifth, 2 for the third and fourth. We can find similar patterns for all the faces.
Let's define "bounce" a move that follows the cycle [1,2,3,3,2,1]. When you apply the bounce move to a face in a certain status, the next status is determined by the next number in the above cycle. For example, if you bounce from 2 (and the previous status was 1), the next status will be 3.
Now a general rule to remember the moves (needs some practice, though):

1. Face #1 oscillates between 0 and 2 every two moves (starting from 2).
2. Face #2 bounces every two moves.
3. Face #3 bounces every six moves.
4. Face #4 bounces every 18 moves.
5. Face #5 bounces every 54 moves.
6. Face #6 bounces every 162 moves.

As you can see, the bounce distances increase with a factor x3 between two consecutive faces.

• I believe the question is:what is the system or theory that governs this sequence of moves, so that you don't just memorize 324 steps but you know a rule like "move odd numbers inward" or "switch largest and smallest numbers" or whatever. I don't see this answer covering that at all – Kate Gregory May 17 '15 at 15:05
• @KateGregory Good point! I'll add more details in my answer asap – leoll2 May 17 '15 at 15:08
• @KateGregory Yes, its the system behind it or formula. As you only have to remember which step you are on, apply the formula so you do not – Tom May 17 '15 at 17:14
• The sequence is the ternary Gray code. en.wikipedia.org/wiki/Gray_code – Jaap Scherphuis Apr 21 '16 at 15:19