A previous note described "fill constrained" muffin problem instances -- instances (m,p) for which every value v in some range low <= v <= high of integer piece sizes is realizable in some person partition for (m,p). It seems empirically true that this constraint ensures a realizable solution to the (m,p) problem. This note describes all "fit constrained" muffin instances (m,p). For these instances, constraints on a single person and muffin partition force a larger spread in the ratio of piece sizes than fill constraints. Recall that I normalize to m >= p and muffin size 1, and that S(m,p) is the smallest piece size required. The fit constrained muffin instances (m,p) and their bounds on S(m,p) are: a). S(m,p) <= 1-m/kp for k > 2, k^2/(2k-1) < m/p < (k+1)/2 b). S(m,p) <= m/(k+1)p for k > 2, k/2 < m/p < (k^2-1)/(2k-1) c). S(m,p) <= 1/3 for 4/3 < m/p < 3/2 Proof: In cases (a) and (b), m/p > 3/2. If an optimal partition requires splitting any muffin into 3 or more pieces then S(m,p) <= 1/3, satisfying the bound. That leaves optimal partitions with muffins split into 2 or 1 pieces. There are no integral m/p in cases (a) and (b), so any solution must split at least one muffin, hence S(m,p) <= 1/2. If we take any such partition with unsplit muffins and halve each of them we don't decrease S(m,p) so we still have an optimal partition. So we may assume each muffin is split into exactly 2 pieces. Sharing 2m pieces among p persons requires that some person receive k or fewer pieces (since k/2 < m/p < (k+1)/2), and the largest piece for that person satisfies L >= (m/p)/k = m/kp > 1/2. L's complement piece in a muffin thus has size S = 1 - L <= 1 - m/kp, establishing case (a). Similarly, some person must receive k+1 or more pieces, and the smallest piece for that person satisfies S <= (m/p)/(k+1) = m/(k+1)p < 1/2. This establishes case (b). For case (c), either some muffin is split in 3 or more pieces and S(m,p) <= 1/3, or every muffin has at most 2 pieces and the argument for case (a) with k=2 gives S(m,p) <= 1-m/2p <= 1-4/(2.3) = 1/3. This reasoning also gives piece size ranges that empirically seem to always realize a solution. If this is always the case, a likely prospect, then these bounds are tight. Piece sizes realizing solutions for these cases, scaled to a range of integer sizes, are: a). pieces kp-m .. m; 2 pieces per muffin, k or k+1 pieces per person b). pieces m .. (k+1)p-m; 2 pieces per muffin, k or k+1 pieces per person c). 3 or 2 pieces per muffin (exact split when 3 pieces), 3 pieces for each person; Reasoning similar to cases (a) and (b) gives the range of piece sizes which appear 2 per muffin. (i). pieces p, m .. 3p-m for 4/3 < m/p <= 7/5 (ii). pieces 2p, 7p-3m .. 3m-p for 7/5 <= m/p < 3/2 (and pieces p, (7p-3m)/2 .. (3m-p)/2 when m+p is even) Some examples: Case (a): S(11,6) = 7/18; pieces 7..11 muffins, scaled size 18: 7.11*6, 8.10*2, 9.9*3 persons, scaled size 33: 11.11.11*2, 7.7.9.10*2, 7.8.9.9*2 Case (b): S(11,7) = 11/28; pieces 11..17 muffins, scaled size 28: 11.17*4, 12.16*2, 13.15*2, 14.14*3 persons, scaled size 44: 11.11.11.11, 17.15.12*2, 17.14.13*2, 16.14.14*2 Case (c.i): S(11,8) = 1/3; pieces 8, 11..13 muffins, scaled size 24: 8.8.8*2, 11.13*6, 12.12*3 persons, scaled size 33: 8.12.13*6, 11.11.11*2 Case (c.ii): S(10,7) = 1/3; pieces 14, 19..23 muffins, scaled size 42: 14.14.14, 19.23*6, 20.22*3, 21.21 persons, scaled size 60: 14.23.23*3, 19.22.22*2, 19.20.21*2