From a7553dfdcd407ffaed23fa945cf5cad32e91b74c Mon Sep 17 00:00:00 2001 From: Simon Tatham Date: Sat, 29 Mar 2014 17:59:25 +0000 Subject: [PATCH] Filled in the observations section. --- template.html | 54 ++++++++++++++++++++++++++++++++++----------------- 1 file changed, 36 insertions(+), 18 deletions(-) diff --git a/template.html b/template.html index ab04962..a5c4951 100644 --- a/template.html +++ b/template.html @@ -430,31 +430,49 @@ apply in that case.

Observations from the data

-FIXME: clear patterns in the m=3 and m=4 columns, and -the bounding proofs seem consistently tight there too, so those cases -may be tractable to establish a complete proof for. -

- -

-FIXME: the diagonal n=m+1 also looks nicely patterned. -We don't have a reliably tight bound there, but it might be worth -trying to prove that diagonal anyway, in the hope that we can +Looking at the above table, there are clear patterns in the m=3 +and m=4 columns. The sequence of fractional values 5/4, 4/3, +11/8, 7/5, 17/12, … shows an obvious regularity if you rewrite it as +3/2 âˆ’ 1/4, 3/2 âˆ’ 1/6, 3/2 âˆ’ 1/8, +3/2 âˆ’ 1/10, 3/2 âˆ’ 1/12, …. And the fractions in +the m=4 column tend upwards to 2 in a similar harmonic series +too. The upper bound proofs seem to be tight everywhere in these +columns as well (assuming the two uncertain cases in the m=4 +column go as expected), so it may be tractable to establish a rigorous +proof completely solving the problem for these two values of m. +

+ +

+There's also a nice pattern down the diagonal n=m+1. Our +current bounding proofs are not reliably tight in that region, but it +might be worth trying anyway to prove that that pattern holds along +the whole diagonal, in the hope that in the process we can discover another useful bounding proof!

-FIXME: the patterns in columns 3 and 4 suggest to me the more -ambitious conjecture that perhaps a similar pattern holds in every -column if you look at cells spaced vertically by m, e.g. all -the points with m=5 and n≡1 mod 5. I don't think we have -enough data here to say anything with confidence about that idea, -though. +The patterns in columns 3 and 4 suggest a more ambitious conjecture to +me. Each of those columns has obviously different behaviour depending +on the value of n mod m (e.g. in the m=3 column +the sequence of fractions is interrupted every third cell because +something obviously different happens when n is a multiple of +3), but if you split up each column into m subcolumns by taking +every mth cell, the pattern in each subcolumn is much simpler, +being either constant or harmonic. So the more ambitious conjecture +isf that perhaps a similarly simple pattern (of some sort) might hold +in every column, if you split it into subcolumns by the value +of n mod m. I don't think we have enough data here to +say anything with confidence about that idea, though.

-FIXME: the case n=29,m=19 is especially intriguing for -the heavy use of denominator 28 in a dissection with min fragment -something/4. +The case n=29,m=19 is especially intriguing for the fact +that the largest denominator used in +the whole dissection is 28, though the +denominator of the minimum fragment is only 4. (And that +dissection was found by the ILP search program, which tries small +denominators first, so I think there probably is no dissection with +the same minimum fragment and less silly other fragment lengths.)

Credits

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