Question 1199921
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Our two figures are not the same; we are moving different directions on the cube.<br>
But how we draw the figure does not change the answer.  We need to move 4 units in each of three directions; it doesn't matter how we name those three directions.  The number of different ways to do that is the number of ways of arranging the symbols AAAABBBBCCCC; that number is<br>
{{{(12!)/((4!)(4!)(4!))=34650}}}<br>
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Since the links to the original figure and the original problem have been lost, I don't know what the original problem was.<br>
The solution above allows paths through the interior of the cube; if the paths must be on the surface of the cube, then the solution above is of course incorrect.<br>