  A concrete computation -- twelve slidings with sixteen tiles -- reveals that
certain commutativity phenomena occur in every double semigroup. This can be
seen as a sort of Eckmann-Hilton argument, but it does not use units. The
result implies in particular that all cancellative double semigroups and all
inverse double semigroups are commutative. Stepping up one dimension, the
result is used to prove that all strictly associative two-fold monoidal
categories (with weak units) are degenerate symmetric. In particular, strictly
associative one-object, one-arrow 3-groupoids (with weak units) cannot realise
all simply-connected homotopy 3-types.
