i'm glad to see that there is still some interest in the muffin/common refinement of partitions problem. some time ago, veit elser gave the upper bound T(m, n) <= 1/(3m) for m < n < 3m/2 , and noted that some of the values he found achieved this bound. i looked at this range more closely and found that i could improve the upper bound as follows. here t denotes a positive integer. if ((3t+1)/(3t))m < n <= ((3t)/(3t-1))m , then T(m, n) <= 1/(3n) . if ((3t+2)/(3t+1))m < n <= ((3t+1)/(3t))m , then T(m, n) <= (t+1)/(2n) - t/(2m) . if ((3t+3)/(3t+2))m < n < ((3t+2)/(3t+1))m , then T(m, n) <= (t+1)/(4m) - t/(4n) . if ((3t+3)/(3t+2))m = n , then T(m, n) <= 1/(3m) . (here, as in previous messages, i am using T(m, n) = maximum c such that there is a common refinement of 1/m + 1/m + ... + 1/m = 1 and of 1/n + 1/n + ... + 1/n = 1 , with all parts >= c .) the first three cases are all improvements over veit's bound. in the fourth case, where there is no improvement, equality is always attained by veit's construction for T(3t+2, 3t+3) , and by obvious extension to T((3t+2)k, (3t+3)k) . however, the above shows that this is the exception, rather than the rule. i find the first case interesting, because i have not seen any cases (with m <= n ) where T(m, n) is strictly less than 1/(3n) . if we allow parts of size 1/(3n) , we may split some of the 1/n 's into 3 parts, and that seems to allow a lot of flexibility. in fact, the only occurrences of T(m, n) = 1/(3n) i've seen are those from the first case above. so i conjecture that equality always holds in the first case. on the other hand, the bounds in the other cases cannot always be attained. they can be attained if n = m+1 , by veit's and moulton's constructions. they can also be attained if n = m+2 by similar constructions. however, if n >= m + 3 , there are cases that require more careful analysis and for which the upper bound can be lowered; the first such case is (13, 16) . when n = m + 3 , it seems that the upper bound can be attained in all cases except when m = 4 mod 9 ; in that case the exact value is T(9k+4, 9k+7) = (9k+5)/(3(9k+4)(9k+7)) for k >= 1 . below are some data i calculated, all by pencil and paper, so subject to a small possibility of error. obviously this amount of data starts to become unwieldy; corrections and/or confirmations are welcome! i believe i have a proof for each of these values, but may have given short shrift to some subcases i thought couldn't give optimal partitions. mike T(2, 3) = 1/6 T(2, 4) = 1/4 T(3, 4) = 1/12 T(2, 5) = 1/10 T(3, 5) = 1/12 T(4, 5) = 3/40 T(2, 6) = 1/6 T(3, 6) = 1/6 T(4, 6) = 1/12 T(5, 6) = 1/15 T(2, 7) = 1/14 T(3, 7) = 5/84 T(4, 7) = 5/84 T(5, 7) = 1/21 T(6, 7) = 1/21 T(2, 8) = 1/8 T(3, 8) = 1/18 T(4, 8) = 1/8 T(5, 8) = 1/20 T(6, 8) = 1/24 T(7, 8) = 5/112 T(2, 9) = 1/18 T(3, 9) = 1/9 T(4, 9) = 7/144 T(5, 9) = 2/45 T(6, 9) = 1/18 T(7, 9) = 5/126 T(8, 9) = 1/24 T(2, 10) = 1/10 T(3, 10) = 2/45 T(4, 10) = 1/20 T(5, 10) = 1/10 T(6, 10) = 1/24 T(7, 10) = 1/30 T(8, 10) = 3/80 T(9, 10) = 1/30 T(2, 11) = 1/22 T(3, 11) = 1/24 T(4, 11) = 9/220 T(5, 11) = 13/330 T(6, 11) = 7/198 T(7, 11) = 1/28 T(8, 11) = 1/33 T(9, 11) = 13/396 T(10, 11) = 7/220 T(2, 12) = 1/12 T(3, 12) = 1/12 T(4, 12) = 1/12 T(5, 12) = 1/30 T(6, 12) = 1/12 T(7, 12) = 1/28 T(8, 12) = 1/24 T(9, 12) = 1/36 T(10, 12) = 1/30 T(11, 12) = 1/33 T(2, 13) = 1/26 T(3, 13) = 11/312 T(4, 13) = 11/312 T(5, 13) = 1/30 T(6, 13) = 1/30 T(7, 13) = 8/273 T(8, 13) = 1/32 T(9, 13) = 1/39 T(10, 13) = 7/260 T(11, 13) = 1/39 T(12, 13) = 1/39 T(2, 14) = 1/14 T(3, 14) = 1/30 T(4, 14) = 1/28 T(5, 14) = 11/350 T(6, 14) = 5/168 T(7, 14) = 1/14 T(8, 14) = 5/168 T(9, 14) = 1/36 T(10, 14) = 1/42 T(11, 14) = 2/77 T(12, 14) = 1/42 T(13, 14) = 9/364 T(2, 15) = 1/30 T(3, 15) = 1/15 T(4, 15) = 13/420 T(5, 15) = 1/15 T(6, 15) = 1/30 T(7, 15) = 1/35 T(8, 15) = 1/40 T(9, 15) = 1/36 T(10, 15) = 1/30 T(11, 15) = 1/45 T(12, 15) = 1/40 T(13, 15) = 3/130 T(14, 15) = 1/42 T(2, 16) = 1/16 T(3, 16) = 7/240 T(4, 16) = 1/16 T(5, 16) = 1/35 T(6, 16) = 1/36 T(7, 16) = 3/112 T(8, 16) = 1/16 T(9, 16) = 11/432 T(10, 16) = 1/40 T(11, 16) = 1/48 T(12, 16) = 1/48 T(13, 16) = 7/312 T(14, 16) = 5/224 T(15, 16) = 1/48 T(2, 17) = 1/34 T(3, 17) = 1/36 T(4, 17) = 15/544 T(5, 17) = 13/510 T(6, 17) = 13/510 T(7, 17) = 11/476 T(8, 17) = 1/40 T(9, 17) = 10/459 T(10, 17) = 2/85 T(11, 17) = 1/44 T(12, 17) = 1/51 T(13, 17) = 9/442 T(14, 17) = 5/238 T(15, 17) = 7/340 T(16, 17) = 11/544 T(2, 18) = 1/18 T(3, 18) = 1/18 T(4, 18) = 1/36 T(5, 18) = 1/40 T(6, 18) = 1/18 T(7, 18) = 1/42 T(8, 18) = 7/288 T(9, 18) = 1/18 T(10, 18) = 1/45 T(11, 18) = 1/44 T(12, 18) = 1/36 T(13, 18) = 1/54 T(14, 18) = 5/252 T(15, 18) = 1/45 T(16, 18) = 1/48 T(17, 18) = 1/51 T(2, 19) = 1/38 T(3, 19) = 17/684 T(4, 19) = 17/684 T(5, 19) = 16/665 T(6, 19) = 1/42 T(7, 19) = 25/1064 T(8, 19) = 13/608 T(9, 19) = 1/45 T(10, 19) = 11/570 T(11, 19) = 9/418 T(12, 19) = 1/48 T(13, 19) = 1/57 T(14, 19) = 1/57 T(15, 19) = 7/380 T(16, 19) = 1/57 T(17, 19) = 1/57 T(18, 19) = 1/57 T(2, 20) = 1/20 T(3, 20) = 1/42 T(4, 20) = 1/20 T(5, 20) = 1/20 T(6, 20) = 1/45 T(7, 20) = 3/140 T(8, 20) = 1/40 T(9, 20) = 1/45 T(10, 20) = 1/20 T(11, 20) = 13/660 T(12, 20) = 1/48 T(13, 20) = 1/52 T(14, 20) = 1/60 T(15, 20) = 1/60 T(16, 20) = 3/160 T(17, 20) = 1/60 T(18, 20) = 1/60 T(19, 20) = 13/760 T(2, 21) = 1/42 T(3, 21) = 1/21 T(4, 21) = 19/840 T(5, 21) = 1/45 T(6, 21) = 1/42 T(7, 21) = 1/21 T(8, 21) = 1/48 T(9, 21) = 5/252 T(10, 21) = 1/50 T(11, 21) = 4/231 T(12, 21) = 5/252 T(13, 21) = 1/52 T(14, 21) = 1/42 T(15, 21) = 1/63 T(16, 21) = 11/672 T(17, 21) = 2/119 T(18, 21) = 1/63 T(19, 21) = 13/798 T(20, 21) = 1/60 T(2, 22) = 1/22 T(3, 22) = 5/231 T(4, 22) = 1/44 T(5, 22) = 9/440 T(6, 22) = 1/48 T(7, 22) = 1/49 T(8, 22) = 9/440 T(9, 22) = 7/396 T(10, 22) = 13/660 T(11, 22) = 1/22 T(12, 22) = 7/396 T(13, 22) = 21/1144 T(14, 22) = 1/56 T(15, 22) = 1/66 T(16, 22) = 1/66 T(17, 22) = 3/187 T(18, 22) = 13/792 T(19, 22) = 13/836 T(20, 22) = 7/440 T(21, 22) = 1/66 T(2, 23) = 1/46 T(3, 23) = 1/48 T(4, 23) = 21/1012 T(5, 23) = 1/50 T(6, 23) = 19/966 T(7, 23) = 19/966 T(8, 23) = 17/920 T(9, 23) = 1/54 T(10, 23) = 17/920 T(11, 23) = 1/55 T(12, 23) = 13/828 T(13, 23) = 53/2990 T(14, 23) = 17/966 T(15, 23) = 1/60 T(16, 23) = 1/69 T(17, 23) = 1/69 T(18, 23) = 19/1242 T(19, 23) = 27/1748 T(20, 23) = 7/460 T(21, 23) = 29/1932 T(22, 23) = 15/1012 T(2, 24) = 1/24 T(3, 24) = 1/24 T(4, 24) = 1/24 T(5, 24) = 7/360 T(6, 24) = 1/24 T(7, 24) = 1/56 T(8, 24) = 1/24 T(9, 24) = 1/54 T(10, 24) = 1/60 T(11, 24) = 19/1056 T(12, 24) = 1/24 T(13, 24) = 5/312 T(14, 24) = 1/56 T(15, 24) = 1/60 T(16, 24) = 1/48 T(17, 24) = 1/72 T(18, 24) = 1/72 T(19, 24) = 9/608 T(20, 24) = 1/60 T(21, 24) = 5/336 T(22, 24) = 1/66 T(23, 24) = 1/69 T(2, 25) = 1/50 T(3, 25) = 23/1200 T(4, 25) = 23/1200 T(5, 25) = 1/25 T(6, 25) = 1/54 T(7, 25) = 1/56 T(8, 25) = 1/56 T(9, 25) = 4/225 T(10, 25) = 1/50 T(11, 25) = 19/1100 T(12, 25) = 1/60 T(13, 25) = 14/975 T(14, 25) = 17/1050 T(15, 25) = 1/60 T(16, 25) = 1/64 T(17, 25) = 1/75 T(18, 25) = 1/75 T(19, 25) = 13/950 T(20, 25) = 3/200 T(21, 25) = 1/75 T(22, 25) = 23/1650 T(23, 25) = 1/75 T(24, 25) = 1/75 T(2, 26) = 1/26 T(3, 26) = 1/54 T(4, 26) = 1/52 T(5, 26) = 1/55 T(6, 26) = 11/624 T(7, 26) = 1/56 T(8, 26) = 11/624 T(9, 26) = 19/1170 T(10, 26) = 1/60 T(11, 26) = 9/572 T(12, 26) = 1/60 T(13, 26) = 1/26 T(14, 26) = 4/273 T(15, 26) = 37/2340 T(16, 26) = 1/64 T(17, 26) = 1/68 T(18, 26) = 1/78 T(19, 26) = 1/78 T(20, 26) = 7/520 T(21, 26) = 11/819 T(22, 26) = 1/78 T(23, 26) = 4/299 T(24, 26) = 1/78 T(25, 26) = 17/1300 T(2, 27) = 1/54 T(3, 27) = 1/27 T(4, 27) = 25/1404 T(5, 27) = 23/1350 T(6, 27) = 1/54 T(7, 27) = 22/1323 T(8, 27) = 7/432 T(9, 27) = 1/27 T(10, 27) = 1/60 T(11, 27) = 17/1188 T(12, 27) = 7/432 T(13, 27) = 1/65 T(14, 27) = 5/378 T(15, 27) = 2/135 T(16, 27) = 13/864 T(17, 27) = 1/68 T(18, 27) = 1/54 T(19, 27) = 1/81 T(20, 27) = 1/81 T(21, 27) = 5/378 T(22, 27) = 4/297 T(23, 27) = 1/81 T(24, 27) = 1/72 T(25, 27) = 17/1350 T(26, 27) = 1/78 T(2, 28) = 1/28 T(3, 28) = 13/756 T(4, 28) = 1/28 T(5, 28) = 1/60 T(6, 28) = 1/60 T(7, 28) = 1/28 T(8, 28) = 1/56 T(9, 28) = 1/63 T(10, 28) = 11/700 T(11, 28) = 1/66 T(12, 28) = 5/336 T(13, 28) = 1/65 T(14, 28) = 1/28 T(15, 28) = 17/1260 T(16, 28) = 5/336 T(17, 28) = 1/68 T(18, 28) = 1/72 T(19, 28) = 1/84 T(20, 28) = 1/84 T(21, 28) = 1/84 T(22, 28) = 1/77 T(23, 28) = 33/2576 T(24, 28) = 1/84 T(25, 28) = 1/84 T(26, 28) = 9/728 T(27, 28) = 1/84 T(2, 29) = 1/58 T(3, 29) = 1/60 T(4, 29) = 27/1624 T(5, 29) = 26/1595 T(6, 29) = 25/1566 T(7, 29) = 1/63 T(8, 29) = 1/64 T(9, 29) = 41/2610 T(10, 29) = 21/1450 T(11, 29) = 1/66 T(12, 29) = 19/1392 T(13, 29) = 45/3016 T(14, 29) = 1/70 T(15, 29) = 16/1305 T(16, 29) = 19/1392 T(17, 29) = 27/1972 T(18, 29) = 1/72 T(19, 29) = 1/76 T(20, 29) = 1/87 T(21, 29) = 1/87 T(22, 29) = 15/1276 T(23, 29) = 49/4002 T(24, 29) = 17/1392 T(25, 29) = 17/1450 T(26, 29) = 1/87 T(27, 29) = 37/3132 T(28, 29) = 19/1624 T(2, 30) = 1/30 T(3, 30) = 1/30 T(4, 30) = 1/60 T(5, 30) = 1/30 T(6, 30) = 1/30 T(7, 30) = 13/840 T(8, 30) = 13/840 T(9, 30) = 2/135 T(10, 30) = 1/30 T(11, 30) = 1/66 T(12, 30) = 1/60 T(13, 30) = 11/780 T(14, 30) = 1/70 T(15, 30) = 1/30 T(16, 30) = 1/80 T(17, 30) = 7/510 T(18, 30) = 1/72 T(19, 30) = 1/76 T(20, 30) = 1/60 T(21, 30) = 1/90 T(22, 30) = 1/90 T(23, 30) = 4/345 T(24, 30) = 1/80 T(25, 30) = 1/75 T(26, 30) = 3/260 T(27, 30) = 1/90 T(28, 30) = 1/84 T(29, 30) = 1/87 T(2, 31) = 1/62 T(3, 31) = 29/1860 T(4, 31) = 29/1860 T(5, 31) = 1/65 T(6, 31) = 1/66 T(7, 31) = 25/1736 T(8, 31) = 25/1736 T(9, 31) = 23/1674 T(10, 31) = 1/70 T(11, 31) = 24/1705 T(12, 31) = 1/72 T(13, 31) = 21/1612 T(14, 31) = 3/217 T(15, 31) = 1/75 T(16, 31) = 17/1488 T(17, 31) = 20/1581 T(18, 31) = 11/837 T(19, 31) = 54/4123 T(20, 31) = 1/80 T(21, 31) = 1/93 T(22, 31) = 1/93 T(23, 31) = 1/93 T(24, 31) = 25/2232 T(25, 31) = 26/2325 T(26, 31) = 1/93 T(27, 31) = 37/3348 T(28, 31) = 19/1736 T(29, 31) = 1/93 T(30, 31) = 1/93 T(2, 32) = 1/32 T(3, 32) = 1/66 T(4, 32) = 1/32 T(5, 32) = 7/480 T(6, 32) = 7/480 T(7, 32) = 1/70 T(8, 32) = 1/32 T(9, 32) = 1/72 T(10, 32) = 1/70 T(11, 32) = 23/1760 T(12, 32) = 1/72 T(13, 32) = 5/416 T(14, 32) = 3/224 T(15, 32) = 1/75 T(16, 32) = 1/32 T(17, 32) = 19/1632 T(18, 32) = 11/864 T(19, 32) = 31/2432 T(20, 32) = 1/80 T(21, 32) = 1/84 T(22, 32) = 1/96 T(23, 32) = 1/96 T(24, 32) = 1/96 T(25, 32) = 11/1000 T(26, 32) = 7/624 T(27, 32) = 1/96 T(28, 32) = 5/448 T(29, 32) = 5/464 T(30, 32) = 1/96 T(31, 32) = 21/1984 T(2, 33) = 1/66 T(3, 33) = 1/33 T(4, 33) = 31/2112 T(5, 33) = 1/70 T(6, 33) = 1/66 T(7, 33) = 1/70 T(8, 33) = 1/72 T(9, 33) = 1/72 T(10, 33) = 3/220 T(11, 33) = 1/33 T(12, 33) = 3/220 T(13, 33) = 1/78 T(14, 33) = 23/1848 T(15, 33) = 13/990 T(16, 33) = 1/80 T(17, 33) = 2/187 T(18, 33) = 7/594 T(19, 33) = 47/3762 T(20, 33) = 49/3960 T(21, 33) = 1/84 T(22, 33) = 1/66 T(23, 33) = 1/99 T(24, 33) = 1/99 T(25, 33) = 17/1650 T(26, 33) = 3/286 T(27, 33) = 13/1188 T(28, 33) = 1/99 T(29, 33) = 10/957 T(30, 33) = 7/660 T(31, 33) = 7/682 T(32, 33) = 1/96 T(2, 34) = 1/34 T(3, 34) = 8/561 T(4, 34) = 1/68 T(5, 34) = 31/2210 T(6, 34) = 1/72 T(7, 34) = 29/2142 T(8, 34) = 15/1088 T(9, 34) = 29/2142 T(10, 34) = 13/1020 T(11, 34) = 1/77 T(12, 34) = 13/1020 T(13, 34) = 1/78 T(14, 34) = 11/952 T(15, 34) = 13/1020 T(16, 34) = 1/80 T(17, 34) = 1/34 T(18, 34) = 5/459 T(19, 34) = 23/1938 T(20, 34) = 1/85 T(21, 34) = 1/84 T(22, 34) = 1/88 T(23, 34) = 1/102 T(24, 34) = 1/102 T(25, 34) = 1/102 T(26, 34) = 9/884 T(27, 34) = 8/765 T(28, 34) = 5/476 T(29, 34) = 1/102 T(30, 34) = 7/680 T(31, 34) = 16/1581 T(32, 34) = 11/1088 T(33, 34) = 1/102 T(2, 35) = 1/70 T(3, 35) = 1/72 T(4, 35) = 33/2380 T(5, 35) = 1/35 T(6, 35) = 31/2310 T(7, 35) = 1/35 T(8, 35) = 29/2240 T(9, 35) = 4/315 T(10, 35) = 1/70 T(11, 35) = 1/77 T(12, 35) = 1/84 T(13, 35) = 64/5005 T(14, 35) = 1/70 T(15, 35) = 1/84 T(16, 35) = 1/80 T(17, 35) = 1/85 T(18, 35) = 19/1890 T(19, 35) = 22/1995 T(20, 35) = 1/84 T(21, 35) = 1/84 T(22, 35) = 1/88 T(23, 35) = 1/92 T(24, 35) = 1/105 T(25, 35) = 1/105 T(26, 35) = 1/105 T(27, 35) = 5/504 T(28, 35) = 3/280 T(29, 35) = 41/4060 T(30, 35) = 1/105 T(31, 35) = 43/4340 T(32, 35) = 11/1120 T(33, 35) = 3/308 T(34, 35) = 23/2380 T(2, 36) = 1/36 T(3, 36) = 1/36 T(4, 36) = 1/36 T(5, 36) = 1/75 T(6, 36) = 1/36 T(7, 36) = 1/77 T(8, 36) = 1/72 T(9, 36) = 1/36 T(10, 36) = 1/80 T(11, 36) = 5/396 T(12, 36) = 1/36 T(13, 36) = 29/2340 T(14, 36) = 1/84 T(15, 36) = 1/90 T(16, 36) = 7/576 T(17, 36) = 1/85 T(18, 36) = 1/36 T(19, 36) = 7/684 T(20, 36) = 1/90 T(21, 36) = 1/84 T(22, 36) = 1/88 T(23, 36) = 1/92 T(24, 36) = 1/72 T(25, 36) = 1/108 T(26, 36) = 1/108 T(27, 36) = 1/108 T(28, 36) = 5/504 T(29, 36) = 5/522 T(30, 36) = 1/90 T(31, 36) = 7/744 T(32, 36) = 1/96 T(33, 36) = 1/99 T(34, 36) = 1/102 T(35, 36) = 1/105 T(2, 37) = 1/74 T(3, 37) = 35/2664 T(4, 37) = 35/2664 T(5, 37) = 33/2590 T(6, 37) = 1/78 T(7, 37) = 33/2590 T(8, 37) = 1/80 T(9, 37) = 1/81 T(10, 37) = 1/80 T(11, 37) = 29/2442 T(12, 37) = 1/84 T(13, 37) = 28/2405 T(14, 37) = 1/84 T(15, 37) = 23/2220 T(16, 37) = 27/2368 T(17, 37) = 59/5032 T(18, 37) = 1/90 T(19, 37) = 20/2109 T(20, 37) = 23/2220 T(21, 37) = 103/9324 T(22, 37) = 9/814 T(23, 37) = 1/92 T(24, 37) = 1/96 T(25, 37) = 1/111 T(26, 37) = 1/111 T(27, 37) = 1/111 T(28, 37) = 19/2072 T(29, 37) = 61/6438 T(30, 37) = 16/1665 T(31, 37) = 1/111 T(32, 37) = 11/1184 T(33, 37) = 1/111 T(34, 37) = 1/111 T(35, 37) = 1/111 T(36, 37) = 1/111