Message #3413

From: phamthihoa4444@gmail.com
Subject: Re: Calculate number of permutation of restricted cube
Date: Thu, 30 Jun 2016 23:04:19 -0700

Why should I visualize it? What I need to prove now is why only 1344 positions are reachable. But is 1344 too large to visualize?

Just like how permutation of 2^4 = (15!/2) * (12^14 * 4) is calculated, hold one corner piece in its place and rotate the rest.

I tried to solve the puzzle by MPUlt (the structure of puzzle file is really difficult to understand, and I have not understand many parts yet) and can solve it intuitively, then I think number of permutation is correct.

Computer brute force given:

• Any 7 cubies in same parity cannot be moved without affecting other pieces. Thus 3, 4, 5, 6 and 7-cycle are all impossible.
  
• Any 4 cubies in same parity cannot be moved if 4 other cubies in that parity is fixed and the other parity can be moved.
  (let a piece be even if all edge-adjacent pieces of it are odd and vice versa)