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This Python program uses a recursive solver to assign each of the words given to one of the possible groupings of cubes. To make it run a bit faster at each recursive step it chooses (one of) the group(s) with the fewest number of positions to fill out. It runs in 330ms.

Solution:The three “words” passing through the centre of the cube are: afd, cfe and efa.The code finds 6 separate solutions, which are the permutations of the X, Y, Z axes.