<h2>Observations from the data</h2>
<p>
-FIXME: clear patterns in the <i>m</i>=3 and <i>m</i>=4 columns, and
-the bounding proofs seem consistently tight there too, so those cases
-may be tractable to establish a complete proof for.
-</p>
-
-<p>
-FIXME: the diagonal <i>n</i>=<i>m</i>+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 <em>can</em>
+Looking at the above table, there are clear patterns in the <i>m</i>=3
+and <i>m</i>=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 − 1/4, 3/2 − 1/6, 3/2 − 1/8,
+3/2 − 1/10, 3/2 − 1/12, …. And the fractions in
+the <i>m</i>=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 <i>m</i>=4
+column go as expected), so it may be tractable to establish a rigorous
+proof completely solving the problem for these two values of <i>m</i>.
+</p>
+
+<p>
+There's also a nice pattern down the diagonal <i>n</i>=<i>m</i>+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 <em>can</em>
discover another useful bounding proof!
</p>
<p>
-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 <i>m</i>, e.g. all
-the points with <i>m</i>=5 and <i>n</i>≡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 <i>n</i> mod <i>m</i> (e.g. in the <i>m</i>=3 column
+the sequence of fractions is interrupted every third cell because
+something obviously different happens when <i>n</i> is a multiple of
+3), but if you split up each column into <i>m</i> subcolumns by taking
+every <i>m</i>th 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 <em>every</em> column, if you split it into subcolumns by the value
+of <i>n</i> mod <i>m</i>. I don't think we have enough data here to
+say anything with confidence about that idea, though.
</p>
<p>
-FIXME: the case <i>n</i>=29,<i>m</i>=19 is especially intriguing for
-the heavy use of denominator 28 in a dissection with min fragment
-something/4.
+The case <i>n</i>=29,<i>m</i>=19 is especially intriguing for the fact
+that the largest denominator used in
+the <a href="#details_29_19">whole dissection</a> is 28, though the
+denominator of the <em>minimum</em> 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.)
</p>
<h2>Credits</h2>
~ianmdlvl / matchsticks-search.git / blobdiff