<div dir="ltr"><div>Thanks, Philip!</div><div><br></div><div>
I approached this problem as an exact set-cover problem. <br><div><br></div><div>Each
(method, leadhead) combo can cover 32 elements from the
set of 672 (1pos, 7pos, 8pos, parity) possibilities. In order to narrow
down the search space, I began the search using all named surprise
methods, but only the 21 in-course leadheads from the 3V calling of
Plain Bob Major, knowing these could be joined into a round block. I
implemented Don Knuth's <a href="https://en.wikipedia.org/wiki/Knuth%27s_Algorithm_X" target="_blank">Dancing Links/Algorithm X</a> in C after Python was being too slow, and after some search-space pruning, it took several days to run finding no solutions. <br></div><div><br></div><div>The largest incomplete subsets it did find were two random 13-lead sets, and a 14-lead set that happened to be all m-type method<font size="1"><font size="2">s. (I can share these as well if there's interest)</font><br></font></div><div><br></div><div>
This led me to add the 21 out-of-course leadheads to the search, but
only look for named m-type methods (adding in TB and Delight). I believe
this ran in an hour or a few, finding 5844 solutions. I then had to filter
the ones that could be joined into a touch of consecutive leads. Finding
none with only 16 and 1678 calls, I added 1238 singles to the mix which
gave me 2040 callings that had methods to back them up. Of these, 31
callings formed round blocks (like Example A), found in the attached file (if attachments are allowed on this list). The others
form 2-, 3-, and 5- part blocks (see examples B and C for some 5-parts):</div><div><br></div><div>A.)
<a href="https://complib.org/composition/56072?accessKey=0b12e98941a37c0650a5536eb3074150e5ce6246" target="_blank">https://complib.org/composition/56072?accessKey=0b12e98941a37c0650a5536eb3074150e5ce6246</a></div><div>B.) <a href="https://complib.org/composition/56084?accessKey=1df6cea0aae7acd3c7570b03a1c0bc13a2f6f8ab" target="_blank">https://complib.org/composition/56084?accessKey=1df6cea0aae7acd3c7570b03a1c0bc13a2f6f8ab</a></div><div> C.)
<a href="https://complib.org/composition/56092?accessKey=73aa71b9b14f23dd9cb48b6fdbe56b735f77de88" target="_blank">https://complib.org/composition/56092?accessKey=73aa71b9b14f23dd9cb48b6fdbe56b735f77de88</a>
</div><div><br></div><div>Several
of the (method, leadhead) combinations have "synonyms", or other
(method, leadhead) pairs with the same (1,7,8,+/-) content. Such leads have interchangeable methods, shown as a list next to each lead in the file. Other sets of leads can be replaced as an ensemble,
which also leads to many method options for a given sequence of calls.
These would be the multiple different touches listed under a given
calling string.<br></div><div><br></div><b>Fewest methods: </b>Of the
21-lead round blocks found, there are 16 method combinations* for
5-method extents (each choice below is independent, except for Moose):<br><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div><span style="font-family:monospace">Moose S<br></span></div><div><span style="font-family:monospace">Brynbuga S/Mark S</span></div><div><span style="font-family:monospace">Norwich S/St Stephen's D</span></div><div><span style="font-family:monospace">Moina S/Tadcaster S</span></div><div><span style="font-family:monospace">Palgrave S/Wendy S</span></div></blockquote><div> </div><div><b>Most methods</b>:
There are 1863 method combinations* that could be chosen for 14-method
extents. I can't print them all here, but all the rounds blocks are in
the attached file.</div><div><br></div><div>Fun fact: every single block I found contains Moose Surprise. There is also a high degree of similarity and overlap between the methods found in each touch.<br></div><div><br></div><div>*probably. If any of these has a call when the tenors cross in 2-3, Ander's skeletons can't be naively applied to those touches.<br></div><div><br></div><div><font size="2"><u><b>Further work:<br></b></u></font></div><div>Some of the multi-part
touches have up to 16 methods. I feel like there might be some way
to use specific part ends instead of the $=123456 call to help cover all
60 in-courses and turn them into extents. <br></div><div><br></div><div>When
filtering out only methods with a single lead-head code, the search
space is small enough to be manageable. I'm running an out-of-course a-type search currently.<br></div><div><br></div><div>I did start an in-course leadhead m-type only search with Roger Bailey's <a href="http://www.ringing.info/all-surprise-major/" target="_blank">list of unnamed methods</a>
in addition to the named ones, but gave up after a few days with
nothing promising found. The search space here is too large to naively
try out-of-course too.<br></div><div><br></div><div>It would be quite
simple to tweak some of the methods in the existing touches to create
new unnamed ones. In particular, you could eliminate some of the singles
by moving them into the first and last changes of a method anywhere
that doesn't affect the tenors, creating "trivial variations". It would
be a bit more effort to create new "synonym" methods for an existing
lead. Or more generally, design a method to fit a particular (1,7,8,+/-) content.<br></div><div><br></div><div>Best,</div><div>Austin</div>
</div></div>