[r-t] Minimus extents
Richard Smith
richard at ex-parrot.com
Fri Sep 22 17:30:18 UTC 2006
Three months ago, I sent a mail here analysing the extents
of minimus that had one bell hunting throughout, and showed
that these could be explained in terms of Q-sets of 12 or 34
singles joining three courses of Original.
http://ex-parrot.com/~richard/r-t/2006/06/001412.html
I've been thinking a bit more about this, and have managed
to extend this reasoning to explain all of the extents of
minimus, even those that at first sight don't have an
obvious Q-set-based explanation. Perhaps more
interestingly, though, it is possible to prove that *all*
extents of minimus can be explained in this way.
Once this is taken into consideration, it is possible to
enumerate (manually, rather than as the result of computer
search) the 38 possible configurations of Q-sets, modulo
translation, reflection, rotation and 'Kent / Oxford'-type
variantion.
Each Q-set configuration works very much like the rules for
a Dixons-like method, and it is simply a case of applying
these to see which produce a viable method. Of the 38
configurations, 29 do actually produce an extent -- and
these correspond to the 29 extents listed on Ander's
website.
http://www.math.ubc.ca/~holroyd/minimus.html
1. Extents in Q-sets
My June 22nd email only looked at extents with a bell plain
hunting, but there's no reason why Q-sets of 12 or 34
singles might not explain any extent where every second
change is a 14 can be explained in these terms.
By having every second change a 14, this means that every
row is adjacent to precisely one -, 12 or 34 change. If we
were only allowing one type of single (whether 12 or 34),
then it is obvious the extents must be explained in terms of
Q-sets of these singles. This is a straightforward
consequence of the Q-set rule. The plain (cross) Q-set is:
1234 1243
2143 2134
and the 34 singled Q-set is:
1234 1243
2134 2143
Clearly either both members of the Q-set have to be plain,
or they both have to be singled.
Ordinarily when there are multiple types of call, the Q-set
rule does not apply -- i.e. the calls do not necessarily
form Q-sets. In the case of 12 and 34 singles, however, we
are helped by a curious fact. The four rows involved in the
-/34 Q-set (that is, the Q-set formed by replacing a -, the
plain, with a 34 single) are the *same* four rows that are
involved in the -/12 Q-set. This means that these four rows
must either all be related by plains (crosses), or all by 12
singles, or all by 34 singles.
According to Ander's list of extents
http://www.math.ubc.ca/~holroyd/min29.txt
there are eight extents that have 14 as every other change,
but that do not have a fixed treble. They are listed below.
#9 2--R O|n|n|u|O
#10 2--- -|n|n|-|n|u|
#11 1P-- I-|-|n|n|-|n|-I
#12 1P-- I-|-|n|u|-|u|-I
#13 1P-- I-|n|u|n|n|u|-I
#24 1--R O|-|n|n|-|u|n|O
#26 1--- -|-|n|n|-|n|-|n|u|n|n|u|
#29 1--- -|n|n|-|n|u|u|n|n|u|n|u|
The third column is the abbrieviated place notation format
that Ander has used on his site:
- : x
| : 14
n : 12
u : 34
=-= : 34x34 or 12x12
=n= : x12x or 34.12.34
=u= : x34x or 12.34.12
I : reflection symmetry point
O : rotation symmetry point
Although it is a bit tedious to verify, the 12s (marked by
'n'), 34s (marked by 'u') and -s are all in Q-sets. As an
example, lets take extent #9 (which is interesting as it is
one of very few single-change minimus extents -- the others
being #1 [Double Court], #2 [Double Canterbury] and #22 [a
palindromic one-part]).
Extent #9
1234 4321 3412 2143
2134 3421 4312 1243
2314 3241 4132 1423
2341 3214 4123 1432
2431 3124 4213 1342
4231 3142 2413 1324
---- ---- ---- ----
4321 3412 2143 1234
This extent has six 12s and six 34s, and if written down,
they can be seen to pair into -/12 Q-sets or -/34 Q-sets
respectively.
-/12 Q-sets | -/34 Q-sets
|
2314 3124 4132 | 1234 2431 4321
2341 3142 4123 | 2134 4231 3421
|
3241 1342 1423 | 2143 4213 3412
3214 1324 1432 | 1243 2413 4312
Once each of these Q-sets has been flipped, the extent
splits into the three courses of Original. (Of course, the
four 14 changes in a course of Original themselves form a
-/14 Q-set, and by flipping these, we could further reduce
the extent to twelve courses of Cross Differential -- the
method with place notation x.x.)
2. Extents with Kent/Oxford Variants
What of the other extents -- those that do not have a 14 as
every other change? Clearly a 14 change cannot be
immediately followed by another 14 change -- it is trivially
false. Similarly, a 14 change cannot be separated from the
next 14 change by more than three other changes -- there are
only four rows available without moving bells across the 2-3
boundary.
Slightly more subtly, there cannot be precisely two other
changes between 14s -- if there were, it would leave one of
the rows with a pair of bells in 1,2 remaining. The only
way that this row could then be reached would be by two
consecutive rows that move bells across the 2-3 boundary; as
14 is the only such change, that would require a repeated
14.
This means that consecutive 14s can only be separated by one
or three other changes. There are six sections of place
notation that can be used to separate 14s by three changes
-- these correspond to the (4-1)! ways of ordering the rows.
Unsurprisingly, it turns out that these six sections of
place notation correspond to the three pairs of Kent/Oxford
variants:
=-= : 34x34 or 12x12
=n= : x12x or 34.12.34
=u= : x34x or 12.34.12
In each case, the difference between the K/O variants is
simply a Q-set being toggled. For example, in 34x34, the
two 34s form a Q-set and can be changed to 12s to give
12x12.
But what of the extents themselves? Can these be explained
in terms of Q-set transformations from, say, Original?
There are 13 extents (modulo K/O variants) to consider.
>From Ander's list these are:
#2 3PGR In=|O [Double Bob]
#3 4P-- I-=|nI [Erin / Stanton]
#5 3P-- In=|-|I [Plain Bob]
#6 2P-- I-|=-=|nI
#8 1PGR I-=|=n=|O
#18 1P-- In=|-|n|=n=|-|I
#19 1P-- In=|=-=|-|=u=|I
#20 1P-- In=|=-=|n|=u=|I
#21 1P-- In=|u|=n=|-|u|I
#22 1P-- In=|u|=n=|n|u|I
#25 1--R O|=-=|n|=u=|u|O
#27 1--- -|=n=|n|-|=-=|=-=|n|=u=|
#28 1--- -|=n=|n|-|=n=|u|=n=|n|u|
It's obvious that Plain Bob can be explained in terms of
Original plus a 14->12 Q-set, and Double Bob as Original
with both a 14->12 and a 14->34 Q-set.
Less obviously, Erin (or Stanton -- they are just K/O
variants of each other), also turns out to be just a few
Q-set transformations away from Original.
Erin Minimus
1234 2413 4321 3142
---- ---- ---- ----
2143 4231 3412 1324
1243 2431 4312 3124
1423 2341 4132 3214
4123 3241 1432 2314
4213 3421 1342 2134
2413 4321 3142 1234
The four cross changes (over the six heads) form one -/14
Q-set. Once these have been removed, we have four blocks of
hunting on three, and these can be converted to three
courses of Original Minimus by changing every 34 to a x
(which is just six -/34 Q-set).
Extent #6 is very similar -- it is just Erin / Stanton with
one -/12 Q-set flipped.
Extent #6 (rotated to emphasise Erin-like nature)
1234 2413 1324 3412
---- ---- ---- ----
2143 4231 3142 4321
1243 2431 1342 3421
1423 2341 1432 3241
4123 s 3214 4132 s 2314
4213 3124 4312 2134
2413 1324 3412 1234
And just when it is beginning to seem that perhaps all
extents of minimus can be built up from Original by
repeatedly flipping Q-set, along comes extent #8...
3. Converting K/O Variants to Q-sets
Extent #8 is quite amusing -- one bell hunts through half of
it, then treble dodges for the remainder, whilst a second
bell treble dodges for the first half, then hunts through
the other half.
Extent #8
1234 1324 2431
2134 3142 4213
2314 3412 4123
3241 3421 1432
2341 * 4312 4132
3214 4321 1423
3124 4231 1243
1342 2413 2143
---- ---- ----
1324 2431 * 1234
In this arrangement, the treble hunts through the first half
extent before swapping with the second which hunts through
the second half, swapping back with the treble at the end.
Given this, it seems likely that the extent is just a
trivial variant of one of the hunt-based extents listed in
my 22 June email. The two cross changes marked with a * are
the only two deviations from pure hunting, and it seems
plausible that these should be part of Q-sets present in the
extent.
In fact, we can prove that this extent cannot be formed from
Original (or equivalently, Cross Differential) by a series
of Q-set transformations. This extent has six 14 changes:
2134 3214 3142 4321 4213 1423
2314 3124 3412 4231 4123 1243
There are three types of Q-set involving 14 changes: -/14
Q-sets containing four 14 changes, and 12/14 and 34/14
Q-sets each containing three 14 changes. There are three
-/14 Q-sets (one per course of Original) and four each of
the 12/14 and 34/14 Q-sets (corresponding to one choice of
bell fixed in 1sts and 4ths place respectively).
A quick inspection of the 14 changes that are present shows
that none of these 3+4+4=11 Q-set are wholly present amongst
the six 14 changes, but that each Q-set is partially
present. This means that there are no Q-set transformations
that will alter the 14 changes present in the extent, and
therefore we cannot transform this extent into Original.
So why does this extent work? In Ander's notation, this
extent is I-=|=n=|O. If we expand the symmetry this
becomes:
=|=u=|=n=|=-=|=n=|=u=|=-
We noted earlier that the K/O variants all have a Q-set in
their outer two changes, and that this explains why there
are two possibilities for each block. For example, the K/O
variant that Ander denotes =-= can be either:
2-2 3-3
: :
1234 1234
1243 2134
2134 1243
2143 2143
: :
In this case, we are simply flipping a 12/34 Q-set.
However, we also noted that the -/12, -/34 and 12/34 Q-sets
all contain the same rows. This suggests that we should be
able to flip the Q-set to have - changes. This is possible,
but fragments the touch into a single - change in place of
the =-= block, plus a -.- round block.
: :
1234 \ : 1243
1243 -------\ 1234 2134
2134 -------/ 2143 ----
2143 / : 1243
: :
If we do this to extent #8, we get a round block of twelve
changes (plus six round blocks of two changes each):
2134 14 :
2314 34 :
3214 14 : 3241 3421 1432
3124 12 : 2341 4312 4132
3142 14 : ---- ---- ----
3412 - : 3241 3421 1432
4321 14 :
4231 12 :
4213 14 : 1342 2413 2143
4123 34 : 1324 2431 1234
1423 14 : ---- ---- ----
1243 - : 1342 2413 2143
---- :
2134 :
This block can be explained in terms of Q-sets if we ignore
the six round blocks of two changes. That this is
necessarily true is not obvious as, in general, the Q-set
rule doesn't apply to touches shorter than an extent. In
this case, though, as alternate changes are 14s, it is
neccessarily true. A 14.c.14 block (for some change, c) has
two rows between the 14 changes; there are then another two
rows (which are either both present or both absent) that
have the same pair of bells in 1-2. These two other rows
must also be related by change c, and the two c's form a
Q-set.
If the change c is absent from the twelve-change block, then
it must be present as one of the c's in a c.c round block as
these are formed from the missing changes between
consecutive 14s. This means that we can necessarily join
the six two-change round blocks into one or more larger
block by changing one of c's in the c.c round block for a
14. (And as the two c's are indistiguishable, it doesn't
matter which.)
With the fragments from extent #8 (shown above), these can
be joined into another twelve-change block:
3241
2341
2431
2413
2143
1234
1324
1342
1432
4132
4312
3421
----
3241
This twelve-change block, together with the one above, cover
the whole extent, and can be explained in terms of two -/12
Q-sets (one when 13 are in 1-2, and the other when 24 are in
1-2), and two -/34 Q-sets (when 14 or 23 are in 1-2).
Flipping these four Q-sets takes us back to Original.
None of this logic is specific to extent #8 and as a result,
any extent of minimus can be converted into a wholly
Q-set-based extent by following this process. Simply flip
the Q-sets in the K/O variants such that they detach to form
two-row round blocks, and use 14s to join all of the two-row
round blocks. The resulting blocks will cover the whole
extent and be comprised of Original transformed by up to six
-/12 or -/34 Q-sets.
(This process applies equally to extents such as Erin /
Stanton which had a simpler Q-set-based descriptions.)
4. The Reverse Transformation
As we now know that every minimus extent can be described in
terms of Original with up to six -/12 or -/34 Q-sets, if we
can enumerate all possible Q-set configurations and reverse
the transformation of K/O blocks above, then we have
produced a list of all extents of minimus.
Reversing the transformation of K/O blocks is straight-
forward. If there is just one round block, there is nothing
to be done and the extent gets no K/O blocks. If there are
several round blocks, then all but one must be split at the
14 changes to get two-row (one change) sections to be
inserted into the remaining round block. Clearly, to
produce and extent, the remaining block must be at least
twelve changes long; so unless there are two twelve-change
blocks, this specifies uniquiely which block to keep. (In
the case of two twelve-change blocks, both possibilities
need trying; in practice, though, both blocks are generally
the same.)
There can only be at most one place to insert each two-row
section -- it will between two rows that have the same pair
of bells on the front. As soon as a section is found that
cannot be inserted, that Q-set configuration can be
discarded -- it cannot be used to produce an extent.
As an example, let's try a 34 single when any of 12, 13 or
23 are in 1-2, and a 12 single when 14, 24 or 34 are in 1-2.
This produces three round blocks, of twelve, six and six
changes respectively.
1234 2413 4321
2134 2431 4312
2314 2341 4132
3214 3241 4123
3124 3421 4213
1324 3412 4231
---- 3142 ----
1234 1342 4321
1432
1423
1243
2143
----
2413
As the second block is the only one with at least twelve
changes, the first and third should be chopped up and
inserted into the second. The first pair of changes from
the first block are 1234 and 2134; these need inserting
between 1243 and 2143 (the other two rows with 1,2 in 1-2),
and these are indeed in the second block. (These rows can
inserted in either order -- and this choice corresponds to
the -34- versus 12.34.12 choice in that K/O block.)
Continuing this, we eventually manage to insert all the rows
from the first and third blocks into the second block to
produce an extent.
Double Bob Minimus (with 4th as hunt)
2413 3421 1432
4231 4312 4123
4213 4321 4132
2431 3412 1423
2341 3142 1243
3214 1324 2134
2314 3124 1234
3241 1342 2143
---- ---- ----
3421 1432 2413
5. Enumeration of Q-set Configuration
The remaining question is how to enumerate the possible
Q-set configurations. There are six Q-sets, each of which
can be in one of three states: -, 12 or 34. This amounts to
3^6 = 729 configurations, however this fails to factor out
configurations that only differ by translation, rotation or
reflection.
The Q-sets can be labeled by the bells in 1-2: 12, 13, 14,
23, 24 and 34. These can be visualised as the six edges of
a tetrahedron, where the vertices correspond to a single
bell:
(1)
/ | \
/ | \
/ |14 \
13/ (4) \12
/ / \ \
/ /34 24\ \
/ / \ \
(3) --------------- (2)
23
Rotations and reflections of the tetrahedron correspond to
translations and vertical reflections (that is, reflections
in a horizontal plane -- the operation under which
palindromic methods are invariant) of the method. (These
permute the vertices of the tetrahedron which corresponds to
starting with an arbitrary row. Depending on whether this
row was previously followed by a 14 change, it may
additionally correspond to a vertical reflection of the
method.)
Modulo rotation and reflection, there are eleven distinct
sets of edges on a tetrahedron. [Is there a good way of
discovering this short of enumerating the posibilities?]
* * *-----* *-----* *-----*
/
/
a) * * b) * * c) * * d) *-----*
*-----* *-----* *-----* *-----*
/ \ \ / \ \ \ / \
/ \ \ / \ \ \ / \
e) * * f) * * g) * * h) * *
*-----* *-----* *-----*
\ \ \ / \ \ \ / \
\ \ \ / \ \ / \ \
i) *-----* j) *-----* k) *-----*
This, however, only deals with a single type of Q-set; we
have two types (12 and 34 changes). Handling this can be
visualised as going through these eleven graphs looking at
the distinct ways of colouring the selected edges red (for
12) and green (for 34).
There is one final element to handle: we have not yet
removed extents (or rather, their Q-set configurations) that
only differ by a reflection in a vertical plane, or
equivalently, rotations. A good start towards this is to
ignore all graphs with more 12s than 34s which only leaves
the case where there are equal numbers of 12s and 34s to
handle (which can only occur when the total number of edges
is even).
To deal with this case, we need to look at what reflection
of the method through a vertical plane does the
representation of a Q-set as an edge on the tetahedron.
Clearly a colour swap occurs as the 12 change becomes a 34
change. But also the bells in 1-2 become those that had
been in 3-4 -- i.e. the complementary pair. Geometrically,
this maps an edge to its opposite edge on the tetrahedron.
This means that under reflection, one configuration does not
necessarily map to differently coloured version of itself --
in particular configuration (e) maps to (f) and vice versa.
(The others all do map to differently coloured versions of
themselves.) This means that (h) and (k) -- the two graphs
with an even number of edges and with (e) and (f) as
subgraphs -- need particularly careful consideration when
there are an equal number of red (12) and green (34) edges.
a-b i) no 12s
c-f i) no 12s
ii) one 12
g i) no 12s
ii) one 12, at end of chain
iii) one 12, in middle of chain
h i) no 12s
ii) one 12, opposite dangling edge
iii) one 12, the dangling edge
iv) one 12, adjacent to dangling edge
v) two 12s, dangling edge plus opposite
vi) two 12s, opposite and adjacent to dangling edge
vii) two 12s, dangling edge plus adjacent
i i) no 12s
ii) one 12
iii) two 12s, adjacent to each other
iv) two 12s, opposite to each other
j i) no 12s
ii) one 12, on perimeter
iii) one 12, diagonal
iv) two 12s, adjacent to each other on different
sides of diagonal
v) two 12s, adjacent to each other on same side of
diagonal
vi) two 12s, opposite to each other
vii) two 12s, diagonal plus another
h i - vii) 12s arranged in configurations (a) - (g)
respectively.
6. The Extents
All that remains is to try out the 38 configurations. The
results of this are given below. The # column gives the
number of the extent in Ander's list -- and it can be seen
that each of the 29 extents occurs there.
Bells in 1-2 | Sizes of | | | Number of
| Partitions | # | Sym | Variants
12 13 14 23 24 34 | | | |
------------------+------------+----+------+---------------
| 8 x3 | - | |
------------------+------------+----+------+---------------
34 | 16 + 8 | 6 | 2P-- | 12.2.16 = 384
------------------+------------+----+------+---------------
34 34 | 24 | 14 | 1P-- | 24.2. 1 = 48
34 12 | 24 | 23 | 1--R | 24.2. 1 = 48
------------------+------------+----+------+---------------
34 34 | 16 + 8 | 3 | 4P-- | 6.2.16 = 192
34 12 | 8 x3 | - | |
------------------+------------+----+------+---------------
34 34 34 | 24 | 4 | 3P-- | 8.2. 1 = 16
34 34 12 | 14 + 10 | 19 | 1P-- | 24.2.32 = 1536
------------------+------------+----+------+---------------
34 34 34 | 18 + 6 | 5 | 3P-- | 8.2. 8 = 128
34 34 12 | 24 | 15 | 1P-- | 24.2. 1 = 48
------------------+------------+----+------+---------------
34 34 34 | 24 | 11 | 1P-- | 24.2. 1 = 48
34 34 12 | 16 + 8 | 27 | 1--- | 24.4.16 = 1536
12 34 34 | 24 | 12 | 1P-- | 24.2. 1 = 48
------------------+------------+----+------+---------------
34 34 34 34 | 18 + 6 | 18 | 1P-- | 24.2. 8 = 384
34 34 34 12 | 24 | 16 | 1P-- | 24.2. 1 = 48
34 34 12 34 | 10 + 8 + 6 | - | |
34 12 34 34 | 24 | 26 | 1--- | 24.4. 1 = 96
34 34 12 12 | 14 + 10 | 20 | 1P-- | 24.2.32 = 1536
34 12 34 12 | 24 | 24 | 1--R | 24.2. 1 = 48
34 12 12 34 | 16 + 8 | 25 | 1--R | 24.2.16 = 768
------------------+------------+----+------+---------------
34 34 34 34 | 12 x2 | - | |
34 34 34 12 | 24 | 10 | 2--- | 12.4. 1 = 48
34 34 12 12 | 24 | 7 | 1PGR | 24.1. 1 = 24
34 12 12 34 | 12 x2 | 8 | 1PGR | 24.1.64 = 1536
------------------+------------+----+------+---------------
34 34 34 34 34 | 12 + 6 x2 | - | |
34 34 34 34 12 | 18 + 6 | 28 | 1--- | 24.4. 8 = 768
12 34 34 34 34 | 12 x2 | - | |
34 34 34 12 12 | 24 | 17 | 1P-- | 24.2. 1 = 48
34 34 12 34 12 | 18 + 6 | 21 | 1P-- | 24.2. 8 = 384
34 12 34 34 12 | 24 | 13 | 1P-- | 24.2. 1 = 48
12 12 34 34 34 | 24 | 29 | 1--- | 24.4. 1 = 96
------------------+------------+----+------+---------------
34 34 34 34 34 34 | 6 x4 | - | |
34 34 34 34 34 12 | 12 + 6 x2 | - | |
34 34 34 34 12 12 | 18 + 6 | 22 | 1P-- | 24.2. 8 = 384
34 34 12 12 34 34 | 12 + 12 | - | |
34 34 34 12 12 12 | 24 | 1 | 3PGR | 8.1. 1 = 8
34 34 12 34 12 12 | 12 + 6 x2 | 2 | 3PGR | 8.1.64 = 512
34 12 34 34 12 12 | 24 | 9 | 2--R | 12.2. 1 = 24
------------------+------------+----+------+---------------
Total 10792
(Note that in each case where the extent partitions into two
round blocks of twelve changes, the blocks are identical and
so it is not necessary to try both ways of combining them.
[I wonder whether there is a good reason why this should be
the case?])
The number of variants of each extent can easily be
calculated from the extents' symmetry and the number of K/O
blocks. The number of distinct translations of an n-part is
24/n, and this is the first factor in the 'number of
variants' column. The number of distinct reflections and
rotations is 1 for maximally symmetry (PGR) extents, 2 for
extents with one symmetry and 4 for asymmetric extents; this
is the second factor in the 'number of variants' column.
The number of K/O variants is given by 2^(12-x/2) where x is
the size of the largest round block. (12-x/2 is the total
number of non-14 changes outside the largest round block,
and hence the number of independent K/O blocks.) This is
the third factor in the 'number of variants' column.
Summing the number of variants of each Q-set configuration
gives the total number of extents of minimus -- 10792 --
which agrees with the results of a computer search.
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