# How many moves are at least needed to change the order of cubes to a arbitrary order? [closed]

We have several cubes that are ordered as in the picture below: What is the smallest number of moves with which we are able to reach every possible order?

Only the order of the cubes in the columns matters, the ordering of the columns on the desk is irrelevant. The height and the number of columns may change.

Valid moves:

• In every step we may take a cube and put it on another cube or on the desk.

1. $11$
2. $14$
3. $17$
4. $23$
5. $25$

My attempt:If we put all cubes on the desk and then bring them together we will need $17$ moves so every case can be made using $17$ moves.I also got a combination that needs at least $16$ moves because it is a test the answer will be $17$ but I need a complete proof.So my question is which combination needs $17$ moves to be made?

• what do you exactly mean by random order? As my understanding there should be no consecutive letters together right? And the does solution has to be the same order with this one ( 5 -3 - 1 -2 )? – Burak Mete Aug 29 '17 at 8:05
• Can't you just have 11 empty spots on the desk and put the cubes there one by one at a randomly chosen still empty location in 11 moves? – Jaap Scherphuis Aug 29 '17 at 8:09
• @Taha Akbari Is A-C taken as a random order or a regular order ? Probably this summarizes Burak Mete's doubt as well.. – Mea Culpa Nay Aug 29 '17 at 8:09
• @BurakMete No I mean a order that every cube can be every where but the order of columns doesn't matter and consecutive letters can be next to each other.And the order doesn't have to be like $(5-3-1-2)$. – Taha Akbari Aug 29 '17 at 8:12
• Either I’m really stupid or the emperor isn’t wearing any clothes.  What do you mean by “order”? – Peregrine Rook Aug 30 '17 at 5:48