Hi Muffinisectors, There's a general upper bound for the muffin problem that, while implicit in other people's posts, has not been explicitly stated, so I record it here. Recall that U(m, n) is the largest size of the smallest part in a common refinement of the partitions m * n and n * m. I'll use square brackets for the floor function. Proposition. For m and n with m not dividing 2n we have U(m, n) <= max( m/3, min( n/([2n/m]+1), m - n/[2n/m] ) ). Proof. Take a partition achieving U(m, n). If some m is divided into at least 3 parts, the smallest of these is at most m/3, and we are done. So we may assume that every m is divided into at most 2 parts, and we need to show that the smallest piece of the refining partition is bounded above by both expressions inside the "min". Since m does not divide n, at least one m is divided into 2 parts, one of which has size at most m/2, so we may split all undivided m's into 2 equal parts without reducing the size of the smallest piece. Then each of the m's is divided into exactly 2 pieces. Since there are n of them, there are exactly 2n pieces in the partition. Now the m n's get divided into 2n pieces, and since m does not divide 2n, some n is divided into at least [2n/m]+1 pieces, and some other n is divided into at most [2n/m] pieces. The smallest of these pieces has size at most n/([2n/m]+1), which proves the first of the two bounds, and the largest of the pieces has size at least n/[2n/m]. But then consider the m that has this last piece as a constituent. It has at least one other piece, and this other piece has size at most m - n/[2n/m], which finishes the proof. QED Now this is interesting mainly for m < n, which we can assume by symmetry, and then you can work out that we always have m/3 < n/([2n/m]+1), and we have m/3 >= m - n/[2n/m] exactly for 4m/3 <= n < 3m/2. Thus we can restate the proposition more conveniently: Corollary. For m < n with m not dividing 2n we have U(m, n) <= m/3 for 4m/3 <= n < 3m/2 U(m, n) <= min( n/([2n/m]+1), m - n/[2n/m] ) otherwise. David P. Moulton