A concrete computation -- twelve slidings with sixteen tiles -- reveals thatcertain commutativity phenomena occur in every double semigroup. This can beseen as a sort of Eckmann-Hilton argument, but it does not use units. Theresult implies in particular that all cancellative double semigroups and allinverse double semigroups are commutative. Stepping up one dimension, theresult is used to prove that all strictly associative two-fold monoidalcategories (with weak units) are degenerate symmetric. In particular, strictlyassociative one-object, one-arrow 3-groupoids (with weak units) cannot realiseall simply-connected homotopy 3-types.
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