Again about the cube Rubik

Again? You may exclaim.
Yes, since the cube Rubik invention in 1974 this plastic enigma became one of the most popular conundrum in the world. Thousands of papers were written proposing many fascinating solution algorithms.
I remember as in my secondary school the cube was an everybody's fad: the pupils spent all their free time in the cube's rotation, made ad-lib break competitions, solved the cube by shortest time, in blind etc.
Well, 36 years later the cube Rubik remains popular. It even seems the faded interest starts growing. At least the new generation takes an interest to the last century invention.
For 3x3x3 cube I recommend the solution at youcandothecube.com (www.youcandothecube.com). For 4x4x4 there are lots of excellent video tutorials at Utube.

Here I'd like to highlight some interesting facts from the cube Rubik's life.

1. The number of all 3x3x3 cube states is: (8!×3^{8−1})×(12!×2^{12−1})/2 = 43 252 003 274 489 856 000. How this number was obtained? The 3x3x3 cube consists of 8 angle elements, 12 edge elements and 6 centres. The centres cannot move relatively each other. So, the angle elements canbe located by 8!×3^8 ways and the edge elements by 12!×2^12 ways. This number counts the rotations of the cube as a solid body as well. So, 6 centres by 4 possible positions counted twice, so in 6x4/2 times the real number of the cube states is less than 8!×3^8×12!×2^12.

2. The most interesting observation is that not all of the possible states canbe reached by simple cube rotation from the original ordered state. Any cube's manipulation affects at least three elements. For example, if any of the cube's corners is 1/3 rotated, the cube cannot be solved. 8 angle elements by 2 their twisted positions give 16. Thus, the number of the cube states obtained from the ordered state at least in 16 times less than 43 252 003 274 489 856 000.

3. The cube maybe solved in blind! This is amazing thing when a human brain is able to keep in mind the current cube's configuration after each rotation. I met such pupils at school.

4. The solution of 4x4x4 cube implies several additional steps leading to 3x3x3 configuration when all edged elements are doubled and centers are in right places. But we may get a configuration which is impossible to solve by 3x3x3 methods without inevitable ungroup the doubled edge elements. In some sence our 4x4x4 cube comes to 3x3x3 configuration being 'turned inside out'.