7.1 I LOVE Pi & Phi

PI & PHI

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In 2010 I created this pi sliding puzzle. 
The game contains 7 discs: 3 blue, 3 gray and 1 white.
The aim of the game is to switch places
between the 3 blue and 3 gray ones.


The game was never commercialized.

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The value of PHI & PI

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Can you prove it for n = 1, 2 en 3?
There is a general proof involving complex analysis!

 

My PIPHI-problem.

 

Let M be the center of the upper face of a cube whose edges have length 1.
This cube is tilted so that it rolls without sliding.
After tilting four times, the point M is on top again.
Show that the total length of the traject of M equals 

HINT. The picture on the left shows you
half of the traject of the point M.

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