On Thu, Jan 8, 2009 at 7:03 PM, Michael Reid <reid@gauss.math.ucf.edu>wrote:
Erich Friedman noticed a nice reciprocity between m and n: if we let S(m,n) denote the size of the smallest piece when m muffins are split among n noshers, then nS(m,n) = mS(n,m).
So, for instance, Rich's observation that S(4,7) is at least 5/21 implies reciprocally that S(7,4) is at least 5/12.
interesting! this suggests to consider instead the function T(m, n) which is the maximum c for which there is a partition of 1 with each part >= c , and that is a simultaneous refinement of 1/m + 1/m + ... + 1/m and also of 1/n + 1/n + ... + 1/n . (as per erich's proof, T(m, n) = S(m, n) / m .)
moreover, this has the obvious generalization to T(n_1, n_2, ... , n_s) , the maximum c for which there is a partition of 1 with each part >= c , and that is a simultaneous refinement of 1/n_i + 1/n_i + ... + 1/n_i for each i .
by the way, i (believe i) can prove that rich's splitting is indeed optimal, and that it is the only way to achieve optimality. before i type in all the details of my proof (and discover the gaps!) can you let me know if you're interested?
Yes! (I'm interested!) --Michael -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.