Indeed, you certainly can't get a longer smallest stick by small perturbations of *that* answer, because one of the 5-sticks is entirely made up of smallest-fragments and so if you lengthen them all by any ε > 0 then that stick overflows. What I was hoping to show was that the converse is also true – that the *only* way in which a dissection can be locally optimal is if some stick is made up entirely of smallest-fragments, by showing that in all other cases you *can* find a system of ε-adjustments that lengthens every smallest-fragment. No luck yet, though!
(In fact, **writinghawk** has proved over on LJ that my dissection for 5-into-7 is globally as well as locally optimal, which is more than I had previously known.)
I've written a thing that looks for integer solutions
Ooh, I'd like to see whatever data it's generated.
Also it can't handle irrationals and so far can do fractions only where all stick-fragments share a denominator
In any rational dissection there must be *some* denominator shared by all fragments (just take the lcm of all denominators), so the latter isn't a problem. And I'm still convinced that irrationals can't appear in any solution unless there's an equally good or better one without them (I have a half-thought-out proof idea involving treating R as a vector space over Q), so I'm not worried about the former either. |