[rt] 147 TDMM
Richard Smith
richard at exparrot.com
Wed Oct 6 01:05:43 UTC 2010
Richard Smith wrote:
> There will be a third (and hopefully shorter) email
> covering extents that can be described in terms of a
> mixture of types of simple splice. For example, extents
> such as the six wrongplace Cambridgeover methods which
> combine a course and a threelead splice (as well as lead
> splices and Parker splices for the 6ths place lead end
> variants).
This is that third email.
But first to correct a typo in the second email. At the end
of the 'MULTIPLE SIXLEAD SPLICES' section, I said:
> [...] That gives a total of 2*6+82+182 = 276 plans.
This should have said 2*6+82+192 = 286 plans. (The 192
terms was correct in the previous table.) The table at the
end of email needs updating accordingly; but this is
repeated (and extended) at the end of this email.
Back to the analysis ...
COMBINING COURSE AND SIXLEAD SPLICES
Each bell pivots once during a course (of a single method)
so it is not possible to combine course and sixlead splices
in a single extent using simple splices. (It might be
possible to do some cunning crosssplice type thing with
suitable methods, though I'm not aware of any. But it would
then no longer be a simple splice and so is beyond the scope
of this calculation).
COMBINING COURSE AND THREELEAD SPLICES
Course splices do combine with threelead splices, as the
extents of the six wrongplace Cambridgeover surprise
methods demonstrate. The table below shows all methods
where XY have a course splice and YZ have a threelead
splice (the fixed bells for which are marked).
X Y Z

Ol Ma Ta (3&5)
Ne Lf Wm (2&5)
Dk/Ox Ms Di (4&5) [see below]
Po Ws Di (2&3)
Ma Ol El (2&4)
Su Du Yo (2&3)
Ey/Do Wl Bo (2&6) [see below]
Ws Po Sa (2&4)
Mu Nw Ak (2&6)
C3 Pn Nm (2&4)
Pn C3 C2 (3&5)
Pv Cx Bn (3&6)
Ce Av Ca (4&5)
Cx Pv Li (2&5)
Lo Cu Cl (2&3)
Nb Cl Cu (2&3)
Cu Lo We (2&4)
(Lead splices and leadend variants have been excluded from
the table for reasons of brevity.)
Lets start with method X and add courses of Y. Clearly we
need at least three courses of Y before we can
exploit the YZ threelead splice. And if we have six
courses of Y, there's no X left and the splice has been
covered elsewhere.
Up to rotation, there are two ways of selecting three
courses to make Y. In one way, the three Y courses share a
coursing pair; in the other way they don't. If the
threelead splice involves a coursing pair (e.g. 3&5 for
MaTa) then the former choice of three courses allows a
single application of the threelead splice and the latter
none; if the threelead splice involves a noncoursing pair
(e.g. 2&5 for LfWm) then it's other choice of three courses
that allows the threelead splice to be applied. Either
way, that gives us one plan (up to rotation).
There is just one way of select four courses of Y. We know
that the two courses of X share two coursing pairs, which
means that 8 = 5*22 coursing pairs have used in the X
leaving two that can be used for the threelead splice. (If
the threelead splice involves noncoursing pairs, change
'coursing' for 'noncoursing' in the preceding sentence.)
That contributes two plans depending on whether we have one
or two applications of the threelead splice.
Finally, five courses of Y which can be chosen in just one
way. Only five coursing pairs are involved in the course of
X leaving five viable threelead splice slots. These are
(2,3), (3,5), (5,6), (6,4), (4,2) if the splice involves a
coursing pair or (2,5), (5,4), (4,3), (3,6), (6,2)
otherwise. Either way, we can label the slots
(a,b), (b,c), (c,d), (d,e), (e,a)
There's one way of choosing one slot, two of choosing two
(together or separate), two of choosing three (all together
or one separate), one of choosing four, and one of choosing
five. That gives seven plans.
So for each set of method (X,Y,Z), we have 10 = 1+2+7 plans.
There are 15 ordinary sets of methods in the table above,
plus a further two with two methods in the X column. In
these, Y coursesplices with both X methods. If we want to
include plans with either (or both) X methods, this gives us
4*1 + 3*2 + 2*7 = 24 plans (up to rotation) for those two
lines.
All in all, that gives us 15*10 + 2*24 = 198 plans.
COMBINING SIXLEAD AND THREELEAD SPLICES
We can also combine threelead and sixlead splices in a
single extent. The table below shows all methods where XY
have a sixlead splice and YZ have a threelead splice (the
fixed bells for which are marked).
X Y Z

Bm Ol El (2&4)
Bp/Cn/Dk/Dn Wl Bo (2&6)
Ki Ma Ta (3&5)
Ma Ki Bo (3&5)
Bh/Bw/By/Cc/Mp Pv Li (2&5)
Ti Tr Qu (2&6)
Cl/Mu Gl Ca (2&3)
Gl/Mu Cl Cu (2&3)
Ak Cz Ww (3&5)
Cz Ak Nw (2&6)
Nw Ww Cz (3&5)
Ww Nw Ak (2&6)
So Pn Nm (2&4)
Fo Li Pv (2&5)
Bn Lo We (2&4)
Lo Bn Cx (3&6)
Cx We Lo (2&4)
We Cx Bn (3&6)
Ne Bo Wl (2&6)
Ne Bo Ki (3&5)
Let's start with an extent of Y and apply the XY sixlead
splice once when bell a pivots. These six leads each rule
out a different threelead splice slot leaving just the four
slots involving bell a: (a,b), (a,c), (a,d), (a,e). That
gives four plans (depending on whether we have 1, 2, 3 or 4
applications of the YZ splice).
If we have a two applications of the XY splice  using
pivots a and b, there's only one threelead slot available:
(a,b). This gives one more plan giving five in total.
There are sixteen sets of methods (X,Y,Z) with a single
method in the X column  that gives 80 = 16*5 plans.
There's a further (4+5+2+2)*4 + (10+15+3+3)*1 = 83
plans from the entries with multiple X methods.
All together, that gives us 163 plans.
OTHER EXTENTS WITH FOUR METHODS
We've now covered all possible simple extents using three
methods. As we know that a simple extent cannot involve
both course and sixlead splices, this leaves four possible
types of threemethod extent:
X (5) Y (5) Z
X (6) Y (6) Z (3) denotes a 3lead splice
X (3) Y (3) Z (5) denotes a course splice
X (5) Y (3) Z (6) denotes a 6lead splice
X (6) Y (3) Z
What about extents with four methods? Quite a lot of these
have been covered too. The course and sixlead splices are
both transitive  that is, if X and Y have a course (or
sixlead) splice, and so do Y and Z, then X and Z do too.
Whenever the XY splice is transitive, the possibility of
multiple X methods has already been considered.
This only leaves a few more possibilities to consider.
W (3) X (5) Y (3) Z
With three courses of X and three courses of Y, if the
WX splice uses a coursing pair and YZ uses a
noncoursing pair then there's exactly one plan with all
four methods. However there are no sets of methods in the
147 that have suitable splices to make this work.
W (5) X (3) Y (3) Z
If we want a single application of YZ splice on (a,b), we
know we can have at most seven applications of XY using:
(a,b), (a,c), (a,d), (a,e), (b,c), (b,d), (b,e). Do these
provide enough X to get a WX course splice? No. Because
we know that the pairs in a course splice are of the form
(p,q), (q,r), (r,s), (s,t), (t,p). So we cannot get four
methods in this way.
W (5) X (3) Y (5) Z
If we one course of W and five of X, then the five pairs
that course / don't course in W can be used in the XY
splice (depending with it uses a noncoursing or coursing
pair). However if we want to add a course of Z we would
need the courses of W and Z not to share any coursing pairs
and that isn't possible. So we cannot get four methods this
way either.
W (3) X (6) Y (3) Z
This cannot work as we know that we need at least 2/5 of the
extent on the threelead splice side of the sixlead splice.
As this has threelead splices on both sides of the sixlead
splice, it cannot work.
W (6) X (3) Y (3) Z
With only six leads of W when bell a pivots, we can get up
to twelve leads of Y whenever bell a is in the fixed
position for the XY splice. However, this leaves no
opportunity for YZ. All four methods have the same
leadend order, and the pivot bell for WX, the two fixed
bells for XY and the two fixed bells for YZ are all
different place bells. If the YZ splice doesn't have a
as a fixed bell, then two of the leads will fall in the W.
If it does have a as a fixed bell then all of the leads are
in the X. Either way, no Z can be included.
W (6) X (3) Y (6) Z
All the methods must have same leadend order which means
WX and YZ have the same fixed (pivot) place bell. If we
ring W when bell a pivots, we can only ring Y when a is
fixed in the threelead splice. Clearly a can't pivot in Z,
but neither can anything else because only those leads with
bell a in the fixed position for YZ are present. So this
doesn't work.
W (3) X (3) Y (3) Z
W (3) X (3) Y

(3)

Z
These plans can both be made to work, but there are no
methods in the 147 that have these particular arrangements
of threelead splices.
W (3) X (3) Y

(5)

Z
This plan cannot work with regular methods. If X has two
threelead splices, then one must involve a coursing pair
and one must involve a noncoursing pair. If we have a
course of Z, then only pairs that do not course in Z are
available for threelead splicing in X
W (3) X (3) Y

(6)

Z
This arrangement of splices is the one that makes a grid
splice work, except that for a regular grid splice, X is an
irregular method and entirely removed. So we know that it
works. Exactly one set of methods in the 147 exists that
has splices in this particular arrangement:
Ki (3&5) Bo (2&6) Wl

(4)

Ne
Let's start with an extent of Bo. We know that we can apply
the BoNe sixlead splice at most twice if we want to be
able to have retain a threelead splice slot for Ki or Wl.
However, we've already counted those plans with only one of
Ki and Wl. Can we get both methods while also including
twelve leads of Ne? Yes. If we ring Ne well bells a or b
pivot, then we can also ring Ki when (a,b) are in 3&5 and Wl
when (a,b) are in 2&6. That's one plan up to rotation.
What about if we only have six leads of Ne, rung when bell a
pivots? That leaves four slots for Ki: (a,b), (a,c), (a,d),
(a,e); and four slots of Wl (with the same fixed bells).
Ignoring Wl, we know there are four ways of choosing Ki, up
to rotation, depending on whether there are 3, 6, 9 or 12
leads of Ki. Adding Wl is more complicated because the
leads of Ki mean the slots are no longer equivalent under
rotation. With one application of BoKi, there are 2+2+2+1
= 7 ways of choosing Wl (depending whether we share the
BoKi fixed pair); with two application of BoKi, there are
2+3+2+1=8 ways of choosing Wl; and by symetry, with three
applications of BoKi there are 7, and with four there are
4.
Finally we need to think about whether chirality is relevant
to any of them. This will only happen if each bell is in
some way unique. The pivot bell in Ne is a, which makes
that unique. If one bell (say e) is not fixed in either Ki
or Wl, that makes that unique. If one bell (b) is fixed in
both Ki and Wl, that can be unique. Which leaves c and d
which can be fixed in Ki and Wl respectively. So the only
plan that splits due to chirality is the plan with two
applications each of BoKi and BoWl, where one pair of
fixed bells is common to the two splices.
That gives 1+7+9+7+4 = 28 plans.
It's worth mentioning in passing that one of these plans
(the one with four applications of BoKi and four of BoWl)
contains no Bo  it has twelve leads of Ki, twelve of Wl
and six of Ne. What's unusual in this case is that we have
a threemethod plan in which none of the methods share a
splice, yet it can be explained in terms of simple
splices by introducing a fourth method. This turns out to
be common with grid splices, though most of the time, the
introduced method (the grid method) is not one of the
methods being considered. For example, with the CmIpBo
grid splice, the grid method is King Edward which is not one
of the 147.
SUMMARY
That brings to an end the analysis of all plans that can be
explained in terms of just simple splices. They can be
grouped as follows:
Single method plans . . . . . . . . . . 75 \
Course splices . . . . . . . . . . . . . 108  See first
Sixlead splices . . . . . . . . . . . . 176  email
Threelead splices . . . . . . . . . . . 798 /
Multiple course splices . . . . . . . . 36 \ See second
Multiple sixlead splices . . . . . . . 286* email
Multiple threelead splices . . . . . . 412 /
Combined course & threelead splices . . 198 \ This
Combined six & threelead splices . . . 163 / email
Other extents with four methods . . . . 28

TOTAL . . . . . . . . . . . . . . . . . 2280
[* = corrected from previous email; see note at top]
It comes as something of a relief that the total of 2280
plans calculated over the three emails in this analysis is
the same as the total number of simple plans counted
automatically by getting a computer to compare plans to each
other, and locating connected components which contain
single method plans.
RAS
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