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simont

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[personal profile] simont Wed 2014-02-19 15:10
there's probably some situation in which you can find a closed loop of those local perturbations that never touch a minimal fragment, in which case you can apply that change with irrational ε to get a partially irrational and equally good answer

In fact, yes, the 7/5 example in my original post admits just such a transformation. Where we previously dissected three 7-sticks as (8/3 + 8/3 + 5/3) and the other two as (7/3 + 7/3 + 7/3), we now do:
  • two lots of (8/3+ε) + (8/3−ε) + (5/3)
  • one unmodified (8/3) + (8/3) + (5/3)
  • two lots of (7/3+ε) + (7/3−ε) + (7/3)
which reassembles as:
  • two lots of (8/3+ε) + (7/3−ε)
  • two lots of (8/3−ε) + (7/3+ε)
  • two lots of (8/3 + 7/3)
  • one lot of (5/3) + (5/3) + (5/3) as before.
And then you can set ε = π/1000 or some such and you have some irrational pieces – but not interestingly irrational.

At a guess, you can probably rule this out by introducing a tie-breaking rule, in which solutions with the same shortest fragment length are now compared by their second shortest, and so on. That'd probably put a stop to frivolous irrationality :-)
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