We consider a variation on Euler Problem 215.

Walls are still built out of 2x1 or 3x1 bricks, but this time is it OK to have "running cracks" of length at most 2, as in the following 9x3 wall:

However, longer running cracks are disallowed.  The following 9x3 wall is not acceptable because it has one running crack of length 3.

There are possible 9x3 walls in which the maximal length of running cracks is 2 or 1.

There are possible 10x4 walls in which the maximal length of running cracks is 2 or 1.

There are possible 11x5 walls in which the maximal length of running cracks is 2 or 1.

Compute the value of .

Hint: consider a graph whose vertices represent pairs of layers, such that an edge represents in acceptable superposition of the three layers , , and .

We consider a variation on Euler Problem 215.

Walls are still built out of 2x1 or 3x1 bricks, but this time is it OK to have "running cracks" of length at most 2, as in the following 9x3 wall:

However, longer running cracks are disallowed.  The following 9x3 wall is not acceptable because it has one running crack of length 3.

There are possible 9x3 walls in which the maximal length of running cracks is 2 or 1.

There are possible 10x4 walls in which the maximal length of running cracks is 2 or 1.

There are possible 11x5 walls in which the maximal length of running cracks is 2 or 1.

Compute the value of .

Hint: consider a graph whose vertices represent pairs of layers, such that an edge represents in acceptable superposition of the three layers , , and .

Consider the 114149 configurations of the 2x2x2 Rubik's cube that require exactly 8 quarter turns to be solved optimaly. 

How many of those have the front face entirely white?

(You should assume the white square with the Rubik's cube logo is always on the front face, as we did in the video.)

Consider the hardest configurations of the 2x2x2 Rubik's cube.  There are 276 configurations that requires 14 quarter turns to be solved optimaly.  How many of those have the front face entirely white?

(You should assume the white square with the Rubik's cube logo is always on the front face, as we did in the video.)

The 2x2x2 Rubik's cube has 3674160 possible configurations.  For 276 of those configurations, the optimal solve requires 14 quarter-turns.  (We consider that a half-turn counts as two quarter-turns.)

If I pick a random configuration out of the 3674160 possible configurations, what is the expected number of quarter-turns required to solve that configuration in the optimal way?  In other words, what is the average number of turns among the optimal solution of all possible configurations?

You can round your result to two decimal places.