[r-t] More from the method-masher - "impossible" triples principles

Alexander E Holroyd holroyd at uw.edu
Fri Jul 22 00:41:36 BST 2022

```A few years ago someone published a remarkable 1-part extent of pure triples with no structure whatever - to all appearances a random string of place notations 1,3,5 and 7.  Moreover this was derived by a fascinating search algorithm involving starting from any extent and successively modifying it, with a key element being reversing sections when falseness is encountered.  I cannot find this original posting, nor can I remember who did this.  Please can someone enlighten me, so that they can have the credit they deserve for this very interesting idea?

Anyway, I have explored extensions of the idea, as I'll explain below.  I propose to call the algorithm the method-masher.  My main discoveries so far are:

1. Any combination of place notations can be used (so long as there are at least 3).
2. It is possible to get a 7-part structure, i.e. a principle whose plain course is the extent.
3. One can seed the process with *any* extent, or 7-part extent accordingly, not necessarily composed of the desired place notations.  (I used https://complib.org/composition/98367).

To illustrate this, here are example principles with place notations 1,3,7 (so pure triples with only 3 place notations), and with 1,345,7 (which have a nice symmetry, and no long places possible), and with 3,147,5 (ditto).

https://complib.org/composition/98370
https://complib.org/composition/98372
https://complib.org/composition/98371

This might be the first known 7-part extent using only place notations 1,3,7, for example.  (Please correct me if not).

Further thoughts and possible limitations:

1. It might be possible to get a symmetric principle this way, although it seems a bit more tricky.
2. There seems to be no way to avoid long places other than choosing place notations that cannot give them, so the approach is of limited use for Minor, for instance.
3. The approach seems only useful for generating methods, or compositions of methods with no internal falseness.  So no use for producing new compositions of Stedman, for example.

Here are the details.

The basic object that we work with is an "extent that does not come round", i.e. a path consisting of all 5040 rows connected by 5039 changes, but with no final change to take it back to the start.  In mathematical terminology this is a Hamiltionian path.  The start and end do not need to be rounds.  If we are starting with a proper extent that does come round, we can get one of these by simply breaking it apart across an arbitrary change, and considering it to start there.

In 7-part world, we instead use a Hamiltonian path in the Schreier graph, i.e. a sequence of 720 rows connected by 719 changes that give the extent when pre-composed by the group of 7 part ends.

Given such a path, from row x to row y say, we choose a place notation to add to the end of it, after the y.  This takes us to a row we already have, say z.  We modify the path so that after the first visit to z we follow the new place notation to y, and then follow the rest of the path backwards until just before we hit z again.  This gives a new path, and we repeat the process.

To avoid going nowhere, the new place notation chosen should be different from the last place notation before y, and also different from the placenotation that was erased at the end of the last step.  But with 3 place notations to choose from this is always possible.

It is possible that z = x, in which case we have found a round block!  If we have done enough mashing we can stop, or if not we can split it in a random place and keep going.

We can start with any extent at all, or any 7-part extent, like https://complib.org/composition/98367.  Eventually all the original place notations get replaced with the ones we want.

With only 3 place notations, there is no choice at a typical step, because 2 of the 3 place notations are disallowed.  I found that occasionally the process would get stuck in an endless loop and never produce a round block.  When that happened I simply backtracked and tried splitting the previous round block in a different place.

cheers, Ander
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