continuing my quest for a "formula" for T(m, n) for the "muffin" problem. here's some further analysis in the case m < n < 3m/2 . as in the case n > 3m/2 , this is still under construction. i previously sent the cases
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) .
david moulton pointed out that something is wrong with these. it seems that i mistakenly included the wrong endpoint in the first two cases. indeed, the four cases should cover the entire range m < n < 3m/2 as t runs through positive integers, without overlap between the cases. i have already used case I to denote the case that m divides 2n , and case II is when n > 3m/2 (and m does not divide 2n ), so case III will be when m < n < 3m/2 , which splits into 4 subcases, as above, but corrected: case III A: ((3t+1)/(3t))m <= n < ((3t)/(3t-1))m here we have the bound T(m, n) <= 1/(3n) , and i believe this is the exact value. (moreover, it seems that T(m, n) > 1/(3n) in all other cases.) the bound T(m, n) <= 1/(3n) for 4m/3 <= n < 3m/n , which is the t = 1 version of case III A, had already been observed by moulton and others. case III B: ((3t+2)/(3t+1))m <= n < ((3t+1)/(3t))m here we have T(m, n) <= (t+1)/(2n) - t/(2m) . this bound cannot always be attained, and i will refine this case below. case III C: ((3t+3)/(3t+2))m < n < ((3t+2)/(3t+1))m here we have the bound T(m, n) <= (t+1)/(4m) - t/(4n) . again, this cannot always be attained, so this case needs refinement as well. case III D: if ((3t+3)/(3t+2))m = n , then T(m, n) = 1/(3m) . this was veit's original bound, and he showed that it is exact in this case. now i refine case III B: divide n by 3(n - m) to get n = 3(n - m)t + u , where n - m < u <= 2(n - m) . (the quotient t is the same integer in the bounds defining case III B.) case III B.1 is if u - (n - m) divides 3(n - m) - u (equivalently, divides 2(n - m) , equivalently, divides 2u ). in this case there is no improvement, and the bound (t+1)/(2n) - t/(2m) can be attained. so now divide 3(n - m) - u by u - (n - m) to get 3(n - m) - u = (u - (n - m))y + z , where 0 < z < u - (n - m) , and compare z(y + 4) to (u - (n - m) - z)(y + 3) . case III B.2 is if z(y + 4) < (u - (n - m) - z)(y + 3) . here we have the improved bound T(m, n) <= ((ty + t + 1)n - t(y + 1)m)/(mn(y + 4)) . case III B.3 is if z(y + 4) > (u - (n - m) - z)(y + 3) . here we have the improved bound T(m, n) <= ((2ty + 3t + y + 3)m - (2ty + 3t + 1)n)/(mn(y + 3)) . case III B.4 is if z(y + 4) = (u - (n - m) - z)(y + 3) . here the bounds in III B.2 and III B.3 coincide, and the common value, ((ty + t + 1)n - t(y + 1)m)/(mn(y + 4)) can be attained. here's my refinement for case III C. i am not entirely satisfied with it, so it is subject to change/reorganization. divide n by 3(n - m) to get n = 3(n - m)t + u , where 2(n - m) < u < 3(n - m) . case III C.1 is if (12/5)(n - m) <= u . at the moment there is no improvement on the bound, although i know it can't always be attained. case III C.2 is if (12/5)(n - m) > u . here we have the improved bound T(m, n) <= ((3t + 3)m - (3t + 1)n)/(3mn) . the first example where the bound in III C.1 is not exact is (31, 38) . it seems that i should split off the case (12/5)(n - m) <= u < (5/2)(n - m) where the bound T(m, n) <= (5t+5)/(4n) - (5t+2)/(4m) holds. i need to double-check this. maybe case III C should be split up in a different way. it's also possible that case III D should be considered as a subcase of case III C. these cases, III A, III B.1,2,3,4, and III C.1,2 give the exact value of T(m, n) in all the examples (where m < n < 3m/2 ) on the long list i previously sent, except for (23, 29) , which goes into case III B.2. so i developed a refinement of that case: case III B.2i is if (u - (n - m) - z)(y + 3) - z(y + 4) >= u - (n - m) - z . here there's no improvement on the bound. case III B.2ii is if (u - (n - m) - z)(y + 3) - z(y + 4) < u - (n - m) - z . here we have the improved bound T(m, n) <= ((4ty + 5t + 2y + 5)/n - (4ty + 5t + 2)/m)/(2(y + 2)) . now (23, 29) goes into case III B.2ii, (with t = 1 and y = 1 ) where the bound is 49/4002 , and this is the exact value of T(23, 29) . presently, i'm not certain if we should expect the bound to always give the exact value in case III B.2i. if not, there is another refinement waiting to be discovered. there's an analogous refinement of case III B.3: case III B.3i is if z(y + 4) - (u - (n - m) - z)(y + 3) >= z . here there's no improvement on the bound. case III B.3ii is if z(y + 4) - (u - (n - m) - z)(y + 3) < z . here we have the improved bound T(m, n) <= ((2ty + 3t + 2)/m - (2ty + 3t + 1)/n)/(2(y + 3)) . some examples: (m, n) = (42, 53) . this is the smallest example that goes into case III B.4. here we have t = 1 and y = 1 (and u = 20 , z = 4 ). the formula gives T(m, n) = 5/742 , and this is attained by the partition 8 * [2 * 15 + 23] + 4 * [15 + 2 * 19] + 10 * [2 * 16 + 21] + 20 * [26 + 27] where i have scaled by 2226 to give integers. i am not giving the rearrangement as a refinement of 53 * 1/53 , since that is automatically determined. i believe that this is the unique optimal partition, and that in case III B.4, there is always a unique optimal partition. (m, n) = (61, 77) goes into case III B.3ii, with t = 1 , u = 29 , y = 1 and z = 6 . the bound in this case becomes 173/37576 , and this is attained by the partition 4 * [2 * 173 + 270] + 4 * [173 + 175 + 268] + 4 * [2 * 175 + 266] + 4 * [174 + 220 + 222] + 2 * [180 + 2 * 218] + 12 * [185 + 187 + 244] + 2 * [2 * 186 + 244] + 12 * [301 + 315] + 4 * [302 + 314] + 12 * [303 + 313] + [2 * 308] where i've scaled by 37576 . (m, n) = (17, 19) goes into case III A with t = 3 . then the bound is T(17, 19) <= 1/57 , and as predicted, is attained by the partition 3 * [17 + 2 * 20] + 2 * [3 * 19] + 6 * [25 + 32] + 6 * [26 + 31] (scaled by 969 ). (m, n) = (32, 37) goes into case III B with t = 2 and u = 7 . here we have u - (n - m) = 2 , which divides 3(n - m) - u = 8 , so we are in case III B.1. the formula gives T(32, 37) = 11/1184 , which is attained by the partition 4 * [2 * 11 + 15] + 4 * [11 + 12 + 14] + 2 * [11 + 2 * 13] + 14 * [16 + 21] + 4 * [17 + 20] + 4 * [18 + 19] . (m, n) = (37, 42) goes into case III C with t = 2 and u = 12 . here we have (12/5)(n - m) = u , so we are in case III C.1. the bound is T(m, n) <= 13/1554 , and this can be attained with the partition 6 * [2 * 13 + 16] + 4 * [3 * 14] + 12 * [18 + 24] + 12 * [19 + 23] + 3 * [2 * 21] . mike