# [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

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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