Sprouts notation
Dan Hoey
response to the discussion earlier this month under the "Game of
Sprouts" topic in geometry-research.
Element 0: Terminology
The number of lines entering a spot is called the spot's "degree".
Degree zero, one, and two spots are called "live" spots; the degree
three spots are "dead".
The lines entering a spot divide the spot into "sites". A spot of
degree zero or one forms a single site. Degree two spots have two
sites and degree three spots have three sites. A degree two spot is
called an "eye spot" if its sites are in separate regions. If both
its sites are in the same region, it is called a "pier spot".
The sites in a region that are connected to each other form a
"boundary." We list the sites of a boundary in "left-hand order,"
the order in which a miniature robot within the region would
encounter them walking forward along the boundary with its left hand
on the boundary. If the boundary surrounds the region, this is
clockwise order. If the region surrounds the boundary, left-hand
order is counterclockwise.
Element 1: Basic position notation
In an N-spot Sprouts game, the initial N spots are numbered 1,2,...,N
and the spots added in the course of the game are numbered starting
at N+1. Each boundary is written as a list of the spot numbers of
its sites in left-hand order, starting at an arbitrary site and
separated by commas. Each region is represented as a list of its
boundaries separated by semicolons. Finally, the region
representations are listed separated by slashes.
1--6---5 Example: This position can be written
| 9 1,6,8,7,4,7,3,9,3,7,8,6,5,6;2/3,9.
8 / \ 2
| \ / Element 2 describes how to abbreviate
4--7------3 this as 1,8,4,9t,8,5;2.
Element 2: Abbreviating position notation
The position notation may be abbreviated by omitting dead spots. If
a degree one spot has all other spots on its boundary deleted it must
be marked with an "o" to distinguish it from a degree zero spot. Any
region with fewer than two live spots may be omitted, but eye spots
listed in only one region must be marked with a "t" to distinguish
them from degree one spots. If the two sites of a pier spot become
adjacent, the pier spot may be listed once, marked with a "t". An
example of the abbreviation appears in the previous example.
It is possible to specify a move in Sprouts by providing the full
position after the move--in fact, the full position notation provides
a complete history of a game. You can see this by using the position
notation to draw the game in pencil, then inking it in one move at a
time. However, it is helpful to be able to specify a move more
concisely, and I describe how to do so below.
Element 3: Basic move notation
A move that connects spot A to spot B, creating spot C in the middle,
is written A-C-B, perhaps with additional notations. By convention,
the lower-numbered endpoint is listed first. The direction of the
move, from A to B, determines one site of C as a "left" and the other
as "right". When listing sites in left-hand order, A is listed after
the left site of C and B is listed after the right site of C. When A
and B are the same spot, this determination is ambiguous, the left
site is taken to be the one accessible to other sites on A's
boundary, if any.
______ Example: The position 1,3,2,4,2,3/2,4 is
(l) / \ created by the moves 1-3-2; 2-4-2. The
1--3--2 (r) 4 (l) first 3 refers to the right site of spot 3,
(r) \______/ and the second 3 refers to the left site.
Element 4: Specifying sites for pier spots
When an endpoint of a move is a pier spot, it is necessary to specify
which site is being connected. This is done by suffixing ".N", where
N is the number of the first live spot appearing after the site in
left-hand order. Just as A is written on the left and B on the right
in the move move A-C-B, the left site is C.A and the right site is
C.B, unless A or B becomes dead after the move. (We avoid using
dead spots to specify sites because they would not appear in the
abbreviated position).
(5.1) Example: After 1-5-2; 3-6-4, there are four ways of
1---5---2 connecting spots 5 and 6, leading to the following
(5.2) four abbreviated positions:
Move 5.1-7-6.3 yields position 1,2,7,3,4,7.
(6.3) Move 5.1-7-6.4 yields position 1,2,7,4,3,7.
3---6---4 Move 5.2-7-6.3 yields position 1,7,3,4,7,2.
(6.4) Move 5.2-7-6.4 yields position 1,7,4,3,7,2.
Element 5: Specifying boundary separations
When a move connects two sites on the same boundary, the region
containing the boundary is split. All other boundaries in the
original region will appear either in the region accessible from the
left site of the new spot or the region accessible from the right
site. This separation is specified by listing the lowest numbered
live spot from each of the boundaries to go into one of the regions.
A list of boundaries accessible to the left site is preceded with
"<", while a list accessible to the right site is preceded with ">".
If there are no other boundaries to separate, the separation is
written "=".
8 7 Example: After 1-6-2; 3-7-4; 1-8-2>; 3-9-4>1
/ \ / \ the position is
1 2 3 5 4
\ / \ / 1-6-2-8;3-9-4-7/1-8-2-6/3-7-4-9;5.
6 9
Element 6: Specifying the region for double-eye-spot moves.
When two eye spots share the same two regions, a move connecting them
must specify in which region the connection is made. Since such eye
spots appear on the same boundary in each region, they separate the
region as in element 5. The separation is also used to determine
which region is used, in one of the following ways:
6a. If the separation normally lists at least one boundary, the
listed boundary is sufficient to specify the region.
6b. If the separation is between at least one boundary on one
side and zero boundaries on the other side, the separation
must specify the side with at least one boundary.
6c. If there are no live boundaries being separated, but there is
at least one other live spot on the boundary being connected,
then the lowest numbered live spot is listed as being
separated, as if it were on a separate boundary, along with
which side the separation is on.
6d. If there are no live spots in the region being separated,
then the separation is written "=".
_ Examples: After 1-3-1> the position is 1,3;2/1,3.
/ \ 2 A move between eye spots 1 and 3 must be written
1---3 1-4-3>2 or 1-4-3= as in elements 6b and 6d,
respectively.
3 In the game 1-4-2; 2-5-3; 1-6-5.2 the position is
| 1,4,2,3,6/1,6,2,4. A move between eye spots 4 and 6
6---5 is written 4-7-6>1 for connection in the region
/ | containing 3, or 4-7-6<1 for the smaller region, as
1--4--2 in element 6c.
Element 7: Equivalent representations for a position
A position representation can be changed in three ways with no
effect on the position represented:
7a: Changing which site is used as the start of a boundary
or boundaries,
7b: Changing the order in which boundaries are listed in a
region or regions, and
7c: Changing the order in which the regions are listed.
There are two other changes that can be made, provided that we do
not need to interpret the notation of moves made in the position:
7d: Permuting the numbers used to denote the spots, and
7e: Reversing the order of all boundaries in a region or
regions.
If a position is changed in these ways, any further moves must be
translated by performing a corresponding permutation to the spot
numbers and exchanging ">" with "<" in the reversed regions (except
where a move connects a point to itself, as in element 3.
I believe these elements are a complete description of a notation
system for Sprouts, but it is quite possible I have overlooked
something. I welcome comments and questions.
Dan Hoey
hao...@aol.com
Danny Purvis
describing positions is ingenious, and the discovery that the original
game notation is deeply flawed is surprising (at least to me) and
valuable. Your recommendations for mending the language seem very
reasonable to me. I hope Dr. Conway will comment.
I am probably still behind on the learning curve, but I have some
preliminary suggestions.
(1) At the moment, this forum seems to be the cutting edge of sprouts
theory, at least on the web. It would be good if everyone interested
could agree on a standard notation.
(2) Since Dr. Conway is a co-inventor of sprouts, his ideas should
carry great weight.
(3) We should be conservative in making changes, since we are trying
to build a tradition.
(4) I prefer keeping the parentheses and reserving dashes for ranges
of consecutive spots in enclosure lists. If you check the oldest
posts on the subject, I think you will see that I originally wanted
dashes for the function you suggest, but Dr. Conway found the
parentheses more pleasing aesthetically. Thus, your basic notation of
a-b-c should remain a(b)c, in my opinion.
(5) The additional notation for docking with pier spots seems to be a
sad necessity. But at least in doing opening analysis, when the games
should be in "canonical form", I think the notation's traditional
preference for low numbers should cause us to draw a move, where
logically equivalent, such that the "b" of "a.b" is the lower of the
two candidate numbers. In cases in which the figure is drawn this
way, I think it should be permissible to drop the ".b" in the
interests of concision. Thus the following two sequences in a three
spot game would be equivalent: (A) "1(4)2 3(5)4.1" and
(B) "1(4)2 3(5)4" .
(6) I favor retaining the square brackets. I don't like terminating a
move with a semicolon since that type of terminator might be confusing
in the context of textual annotations. I also think that skillful
players will eventually be interested mostly in positions with larger
numbers of boundaries. Longer lists look better tucked away in square
brackets in my opinion. I suggest "[list]" instead of "<list;" and
"[|list]" instead of ">list;". (Dr. Conway made some mention of using
"|" inside the square brackets.) And I think that in opening
analysis, the analyst should prefer drawing figures such that the
"[list]" is more concise than the "[|list]".
(7) In the case of a move connecting two eye spots sharing regions, I
favor retaining the use of the "@" symbol to indicate (where not
obvious from normal "enclosing" considerations) the location of the
new spot. I think this language feature is more colorful, simpler, and
more concise than the rules you give. Thus in an eight spot game
after 1(9)1[], I prefer 1(10@2)9 to 1(10)9[2-8].
Well, I need to think about it some more. I would not be amazed to
learn that my remarks reveal some big misunderstanding on my part.
Danny Purvis
I propose the following simplified notation.
(1) The notation will consist of the original notation (the notation
as it existed before Dan Hoey's recent bombshell) with the additions
explained in (2) and (3) below.
(2) Move "f(g)h" is possibly modified if f or g is a pier spot before
the move is played. Let x be former pier spot f or g. A tiny robot R
stands on g and walks towards x. Iff R finds that x's left hand
neighbor is higher numbered than x's right hand neighbor, an
exclamation point is inserted between "x" and "(g)" in the move
notation.
Example: In a five spot game, after 1(6)2 3(7)4, there are four ways
to connect pier spots 6 and 7. These are 6(8)7, 6!(8)7, 6(8)!7, and
6!(8)!7.
(3) In move "f(g)h[list]", iff f <> g and f and h are in
counterclockwise order in relation to [list], then the move notation
is changed to "f(g)h[!list]".
*******************************
The "!" might be called the "Hoey exclam".
In both cases, the Hoey exclam might be thought of as indicating a
disappointment of clockwise expectations.
Danny Purvis
>(3) In move "f(g)h[list]", iff f <> g and f and h are in
>counterclockwise order in relation to [list], then the move notation
>is changed to "f(g)h[!list]".
I'll try again (and probably still mess it up):
(3) For move notation "f(g)h[list]", if f <> h and if an arrow drawn
from f through g to h indicates a counterclockwise rotation around the
boundaries specified by "[list]", then the move notation is changed to
"f(g)h![list]". (It should be noted that f < h, because of the
preference for low numbers.)