Let
be a vector in
, where each component is the value on a face of the cube. It is easy to define the linear transformation
that represents a turn. The vector space
can be decomposed into three
-invariant subspaces:
= {
| all faces have the same value},
={
| the sum of all faces is zero and opposite faces have the same value}, and
={
| the sum of all faces is zero and opposite faces also sum to zero}. Examining the action of
on each subspace makes the reasons for the convergence to the average value clear.
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