Spliced treble-dodging minor - 3 - Revision history

GACJ: Formatting improvements and removal of corrections to part 2.

2010-11-06T18:52:45Z

Formatting improvements and removal of corrections to part 2.

GACJ at 18:16, 6 November 2010

2010-11-06T18:16:31Z

GACJ: moved Spliced treble-dodging minor plans 3 to Spliced treble-dodging minor - 3

2010-10-28T21:30:45Z

moved Spliced treble-dodging minor plans 3 to Spliced treble-dodging minor - 3

Holroyd: Created page with 'Richard Smith richard at ex-parrot.com Wed Oct 6 02:05:43 BST 2010 This is that third email. But first to correct a typo in the second email. At the end of the 'MULTIPLE SIX-…'

2010-10-28T06:47:50Z

Created page with 'Richard Smith richard at ex-parrot.com Wed Oct 6 02:05:43 BST 2010 This is that third email. But first to correct a typo in the second email. At the end of the 'MULTIPLE SIX-…'

New page

Richard Smith richard at ex-parrot.com
Wed Oct 6 02:05:43 BST 2010

This is that third email.

But first to correct a typo in the second email. At the end
of the 'MULTIPLE SIX-LEAD 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 SIX-LEAD SPLICES

Each bell pivots once during a course (of a single method)
so it is not possible to combine course and six-lead splices
in a single extent using simple splices. (It might be
possible to do some cunning cross-splice 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 THREE-LEAD SPLICES

Course splices do combine with three-lead splices, as the
extents of the six wrong-place Cambridge-over surprise
methods demonstrate. The table below shows all methods
where X-Y have a course splice and Y-Z have a three-lead
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 lead-end 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 Y-Z three-lead 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
three-lead splice involves a coursing pair (e.g. 3&5 for
Ma-Ta) then the former choice of three courses allows a
single application of the three-lead splice and the latter
none; if the three-lead splice involves a non-coursing pair
(e.g. 2&5 for Lf-Wm) then it's other choice of three courses
that allows the three-lead 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*2-2 coursing pairs have used in the X
leaving two that can be used for the three-lead splice. (If
the three-lead splice involves non-coursing pairs, change
'coursing' for 'non-coursing' in the preceding sentence.)
That contributes two plans depending on whether we have one
or two applications of the three-lead 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 three-lead 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 course-splices 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 SIX-LEAD AND THREE-LEAD SPLICES

We can also combine three-lead and six-lead splices in a
single extent. The table below shows all methods where X-Y
have a six-lead splice and Y-Z have a three-lead 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 X-Y six-lead
splice once when bell a pivots. These six leads each rule
out a different three-lead 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 Y-Z splice).

If we have a two applications of the X-Y splice -- using
pivots a and b, there's only one three-lead 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 six-lead splices, this leaves four possible
types of three-method extent:

X --(5)-- Y --(5)-- Z
X --(6)-- Y --(6)-- Z --(3)-- denotes a 3-lead splice
X --(3)-- Y --(3)-- Z --(5)-- denotes a course splice
X --(5)-- Y --(3)-- Z --(6)-- denotes a 6-lead splice
X --(6)-- Y --(3)-- Z

What about extents with four methods? Quite a lot of these
have been covered too. The course and six-lead splices are
both transitive -- that is, if X and Y have a course (or
six-lead) splice, and so do Y and Z, then X and Z do too.
Whenever the X-Y 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
W-X splice uses a coursing pair and Y-Z uses a
non-coursing 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 Y-Z splice on (a,b), we
know we can have at most seven applications of X-Y using:
(a,b), (a,c), (a,d), (a,e), (b,c), (b,d), (b,e). Do these
provide enough X to get a W-X 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 X-Y
splice (depending with it uses a non-coursing 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 three-lead splice side of the six-lead splice.
As this has three-lead splices on both sides of the six-lead
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 X-Y splice. However, this leaves no
opportunity for Y-Z. All four methods have the same
lead-end order, and the pivot bell for W-X, the two fixed
bells for X-Y and the two fixed bells for Y-Z are all
different place bells. If the Y-Z 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 lead-end order which means
W-X and Y-Z 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 three-lead 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 Y-Z 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 three-lead splices.

W --(3)-- X --(3)-- Y
|
(5)
|
Z

This plan cannot work with regular methods. If X has two
three-lead splices, then one must involve a coursing pair
and one must involve a non-coursing pair. If we have a
course of Z, then only pairs that do not course in Z are
available for three-lead 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 Bo-Ne six-lead splice at most twice if we want to be
able to have retain a three-lead 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 Bo-Ki, there are 2+2+2+1
= 7 ways of choosing Wl (depending whether we share the
Bo-Ki fixed pair); with two application of Bo-Ki, there are
2+3+2+1=8 ways of choosing Wl; and by symetry, with three
applications of Bo-Ki 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 Bo-Ki and Bo-Wl, 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 Bo-Ki and four of Bo-Wl)
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 three-method 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 Cm-Ip-Bo
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
Six-lead splices . . . . . . . . . . . . 176 | email
Three-lead splices . . . . . . . . . . . 798 /
Multiple course splices . . . . . . . . 36 \ See second
Multiple six-lead splices . . . . . . . 286*| email
Multiple three-lead splices . . . . . . 412 /
Combined course & three-lead splices . . 198 \ This
Combined six- & three-lead 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

← Older revision		Revision as of 18:16, 6 November 2010
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		+	[[Spliced treble-dodging minor - 2\|Plans 2]]
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		+	[[Spliced treble-dodging minor - 4\|Plans 4]]
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		+	[[Spliced treble-dodging minor - 5\|Plans 5]]
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		+	[[Spliced treble-dodging minor - 6\|Plans 6]]
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	Richard Smith richard at ex-parrot.com		Richard Smith richard at ex-parrot.com
	Wed Oct 6 02:05:43 BST 2010		Wed Oct 6 02:05:43 BST 2010

@@ Line 15: / Line 15: @@
 Richard Smith richard at ex-parrot.com
 Wed Oct 6 02:05:43 BST 2010
 Back to the analysis ...
+====COMBINING COURSE AND SIX-LEAD SPLICES====
 Each bell pivots once during a course (of a single method)
@@ Line 42: / Line 28: @@
 of this calculation).
+====COMBINING COURSE AND THREE-LEAD SPLICES====
 Course splices do combine with three-lead splices, as the
@@ Line 50: / Line 35: @@
 where X-Y have a course splice and Y-Z have a three-lead
 splice (the fixed bells for which are marked).
+<pre>
     X      Y   Z
     ----------------------
@@ Line 73: / Line 58: @@
     Nb     Cl  Cu    (2&3)
     Cu     Lo  We    (2&4)
+</pre>
 (Lead splices and lead-end variants have been excluded from
 the table for reasons of brevity.)
@@ Line 128: / Line 113: @@
 All in all, that gives us 15*10 + 2*24 = 198 plans.
+====COMBINING SIX-LEAD AND THREE-LEAD SPLICES====
 We can also combine three-lead and six-lead splices in a
@@ Line 135: / Line 119: @@
 have a six-lead splice and Y-Z have a three-lead splice (the
 fixed bells for which are marked).
+<pre>
     X               Y    Z
     --------------------------------
@@ Line 161: / Line 145: @@
     Ne              Bo   Wl    (2&6)
     Ne              Bo   Ki    (3&5)
+</pre>
 Let's start with an extent of Y and apply the X-Y six-lead
 splice once when bell a pivots.  These six leads each rule
@@ Line 180: / Line 164: @@
 All together, that gives us 163 plans.
+====OTHER EXTENTS WITH FOUR METHODS====
 We've now covered all possible simple extents using three
@@ Line 260: / Line 243: @@
 bell a in the fixed position for Y-Z are present.  So this
 doesn't work.
+<pre>
     W --(3)-- X --(3)-- Y --(3)-- Z
@@ Line 268: / Line 251: @@
+              |
+              Z
+</pre>
 These plans can both be made to work, but there are no
 methods in the 147 that have these particular arrangements
@@ Line 353: / Line 336: @@
 of the 147.
+====SUMMARY====
 That brings to an end the analysis of all plans that can be
@@ Line 365: / Line 347: @@
     Three-lead splices . . . . . . . . . . .  798 /
     Multiple course splices  . . . . . . . .   36 \ See second
-    Multiple six-lead splices  . . . . . . .  286*|   email
+    Multiple six-lead splices  . . . . . . .  286 |   email
     Multiple three-lead splices  . . . . . .  412 /
     Combined course & three-lead splices . .  198 \ This
@@ Line 372: / Line 354: @@
     ---------------------------------------------
     TOTAL  . . . . . . . . . . . . . . . . . 2280
 It comes as something of a relief that the total of 2280

← Older revision	Revision as of 21:30, 28 October 2010
(No difference)