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[personal profile] simont Fri 2014-02-21 13:46
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, [livejournal.com profile] 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.
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