| 1. | Any permutation of the corners is possible, including odd permutations.
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| 2. | Any permutation of the wedge pieces is possible, including even and odd permutations.
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| 3. | Any permutation of the corners is possible, including odd permutations, giving 8 ! possible arrangements.
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| 4. | Any permutation of the outer edges is possible, including odd permutations, giving 24 ! arrangements.
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| 5. | In terms of permutations the two group elements of are the set of even permutations and the set of odd permutations.
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| 6. | In three dimensions only, the cyclic permutations of are all even permutations, similarly the anticyclic permutations are all odd permutations.
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| 7. | Any permutation of the edges is possible, including odd permutations, giving 24 ! arrangements, independently of the corners or centres.
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| 8. | Taking the odd permutations of the above coordinates with an even number of plus signs gives another form, the enantiomorph of the other one.
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| 9. | An odd permutation followed by an odd permutation will represent an overall even permutation ( adding two odd numbers always returns an even number ).
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| 10. | An odd permutation followed by an odd permutation will represent an overall even permutation ( adding two odd numbers always returns an even number ).
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