Muffin problem instances seem to fall into two groups, "fit constrained" and "fill constrained". This note describes the cases, motivates how they arise, and shows how to find fill constrained bounds which are good candidates for optimal. (19,11) is a fill-constrained example, with S(19,11) = 9/22 (recall that I normalize to muffin size 1 and m muffins >= p people). Piece sizes for the (19,11) problem, scaled to integer values, are 9..13. Muffin partitions (scaled size 22) are: 9.13*12, 10.12*6, 11.11 Person partititions (scaled size 38) are: 13.13.12*6, 9.9.10.10*3, 9.9.9.11*2 This example, using piece sizes 9..13, requires each piece size in the range to be used. Note that piece sizes 11 and 12 can appear in a person partition in a unique way: 4 piece partition 9.9.9.11 and 3 piece partition 13.13.12. For 3/2 < m/p < 2, there are optimal partitions with exactly 2 pieces per muffin, 3 or 4 pieces per person, and at least one person with 3 pieces and one with 4 pieces. The "muffin fill constraint" for a range S..L of integer piece sizes is that for every integer k in S..L, a person partition containing k can be constructed. Numerically this is: Strong Fill Constraint for 3/2 < m/p < 2: For range S..L of integer piece sizes and scaled person size Z, (Z-3S) + 1 = Z-2L. Note that Z-3S is the largest piece that can occur in a 4 piece partition and Z-2L is the smallest piece that can appear in a 3 piece partion. The Strong Fill Constraint is that these sizes differ by 1 with the former size smaller. There as also a Weak Fill Constraint, which constrains these two sizes to be equal. Weak Fill Constraint for 3/2 < m/p < 2: Z-3S = Z-2L. For 8/5 <= m/p <= 9/5, it appears that any range of sizes S..L satisfying the strong or weak fill constraint yields a muffin partition and a lower bound S(m,p) >= S/(S+L) on the smallest muffin piece. The Weak Fill Constraint for all 3/2 < m/p < 2 is L/S = 3/2, and apparently S(m,p) >= 2/5 for all m/p in this range. Instances that satisfy the Strong Fill Constraint have S(m,p) > 2/5 when a strong fill partition is realizable. To satisfy the Strong Fill Constraint for m/p (given in lowest terms), look for a continued fraction covergents L/S to 3/2 from below for which p divides S+L, i.e., (4+3k)/(3+2k) for the least k at which p divides 7+5k if such a k exists. I believe such k exists iff gcd(p,5) = 1. I now give several fill constrained examples for 8/5 <= m/p <= 9/5, listing the piece size range but not the actual partition. S(12/7) = 3/7, pieces 3..4 S(7/4) = S(5/3) = 5/12, pieces 5..7 S(16/9) >= 11/27, pieces 11..16 S(19/11) >= 9/22, pieces 9..13 S(17/10) >= 2/5, pieces 4..6 The S(17/10) bound is from thw Weak Fill Constraint, and I don't think the Strong Fill Constraint is satisfiable for 17/10. The fill constrained intervals for m/p > 1 are (k^2-1)/(2k-1) < m/p < k^2/(2k-1) for k > 1 The Strong Fill Constraint in each such interval is (Z - kS) + 1 = Z - (k-1)L. I won't cover all "fit constrained" cases in this note, but I give one example: S(11/6) = 7/18, with piece sizes 7..11, muffin partitions (scaled size 18) 7.11*6, 8.10*2, 9.9*3 and person partitions (scaled size 33) 11.11.11*2, 7.7.9.10*2, 7.8.9.9*2 For 9/5 <= m/p < 2 we may take each muffin to have exactly 2 pieces, since if any muffin is not split, we may create a new partition with the same size smallest piece by halving each unsplit muffin. Since each muffin has 2 pieces, some person partition has 3 or fewer piece, and that partition has some piece of size L >= (m/p)/3 > 1/2. This piece must appear in some muffin partition with smallest piece S <= 1-m/(3p) < 1/2. This reasoning gives L/S >= m/(3p-m) and S(m,p) <= 1-m/(3p) for all 9/5 <= m/p < 2, including our S(11/6) = 7/18 example. Note that the 11/6 example has all 3 piece person partitions composed of only the largest piece, and this largest piece forces a smaller piece that that required for the fill constraint. In fill constraint notation, z - 3S >= Z - 2L in fit contstrained cases. Having a larger range of piece sizes for fit constrained cases gives more flexibility in filling partitions, so these cases should be even easier to realize as partitions than fill constrained cases. If fit constrained cases can always be realized as partitions then the fit constraint bound is tight.