The middle levels problem is to find a Hamilton cycle in the middle levels, M 2k+1 , of the Hasse diagram of B 2k+1 (the partially-ordered set of subsets of a 2k + 1-element set ordered by inclusion). Previously, the best known, from [I. Shields, C.D. Savage, A Hamilton path heuristic with applications to the middle two levels problem, in: ], was that M 2k+1 is Hamiltonian for all positive k through k = 15. In this note we announce that M 33 and M 35 have Hamilton cycles. The result was achieved by an algorithmic improvement that made it possible to find a Hamilton path in a reduced graph (of complementary necklace pairs) having 129,644,790 vertices, using a 64-bit personal computer.

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