Naccel: a[N
accel]erated Pipcount
by
Nack Ballard
Count
-- On Me?
"Twelve plus
fifteen plus six in my board, that's thirty-three, plus sixteen for the
two on my
eight point, that's forty-nine, plus nine equals fifty-eight, and fourteen
equals seventy-two, plus three on the... what IS that, the 'twenty-two' point,
is sixty-six, added to.... Oh no, what was it again?..." I mumbled aloud.
"I wasn't
really paying attention... sixty-six?" my doubles partner suggested
dubiously. "You had just counted
the checker on their eleven point", he added helpfully.
"Well, yeah,
thanks, I know that... but it's the ANCHOR that is sixty-six. I forgot the running count I was
supposed to add it TO..." Here I go
again, I thought... now I'll have to start all over again for the tie-breaking
count. "What were my first two
tries?"
"Hmm....they
differed from each other by eleven... I remember that..." his voice faded as
quickly as his grin.
I could
already feel my equity shrinking.
"Well, let's start again with what we know. What did we get for the count on their
side, 130-what?"
"I know it
was my JOB to remember the counts", he savored the word like a sour plum. "But when you asked me about your last
pipcount, it all went out of my head."
"I wasn't
asking you, I was asking myself".
Hmm, that didn't come out right.
"Well, my
ability to distinguish rhetoric isn't what it used to be, nor is my memory. I can't even recall why I paid both
halves of our entry fee."
I was no
longer sure myself of the reason. I
wanted to quip back on his misuse of the word "rhetoric", but someone had to
pull the team back together.
"Okay,
LOOK..." Pretending I was still in
charge was the only card I had left to play. "You count Red and I'll count Black;
we'll do it really carefully this time, and whatever we get, we'll just go with
THAT."
... Sound
familiar? Well, maybe your presence
of mind is not quite as absent as mine, but might not such a calamity occur if
you find yourself deprived of enough sleep, or otherwise lacking
focus?
Counting
pips costs time, and drains our reserve of energy. It is important to be able to stay
intent on the positional considerations, which, if assessed correctly, will more
likely guide us to correct plays and cube decisions.
------------------------------------------------------------
Following
a Dream
With the
good of the human race in mind, then
-- well okay, I admit it was because I wearied of my own incompetence -- I
actually dreamt (literally) a new method of pipcount. In my dream, I was playing on Gamesgrid,
clicking on the pipcount button (that wonderful online crutch), when out popped
a cyberboard with the four quadrants taking turns bulging out at
me.
The next
morning, while driving, I mused how easy counting the race would be if all I had
to do was count FOUR BIG points instead of 24 little ones. No more two-digit numbers to add or
multiply.
Thinking
further, it seemed as if this simple scheme would indeed give a good
approximation. Beginning players
could benefit by being taught to initially adopt a Quadrant Count, to get
a feel for how much they were ahead or behind.
Then it hit
me. An expert player could use this
absurdly simple counting method, too.
The only other step would be to count the remaining pips in each
quadrant!
When I
arrived home, I set up some backgammon positions. To the checkers in my home board I
assigned a quandrant value of 1, to those in my outer tables 2 or 3, and to my
back checkers 4. Counting the total
quadrants was as easy as counting a bearoff in which my checkers were all on the
4, 3, 2 and 1 points.
To do full
pipcounts, I altered the designations of the quadrants (from 1, 2, 3, 4) to
0, 1, 2, and 3, in order to allow residual pips to be ADDED instead of
subtracted. (A checker on the "22
point" needs to travel THREE (not four) quadrants in order to bear IN, and then
4 more pips to bear OFF). Hmm...
even easier!
I started
leafing through any backgammon literature I could find which had pipcounts
accompanying the diagrams. At
first, I made a lot of silly mistakes, overlooking leftover pips. But, as I kept practicing, and gradually
streamlined the method (in particular, after introducing "squads"), I was
astounded to find myself rattling off correct pipcounts in a matter of seconds.
A wave of
euphoria, bottled up by 25 years of slavish pipcount, washed over me. I shared the details of my discovery
with a friend, Ulf Wostner, who mirrored my enthusiasm by offering to set up a
website (see the end of this article) to teach this new method of counting
pips.
It was Ulf,
in fact, who convinced me to dub this system "NACCEL". I have to admit, I like it. As I have no children, perhaps I imagine
this appellation as some alternate way of spreading my genes. I think it is only fair to clarify that
I attribute neither my psychological imbalance, nor my need to compensate, to
any lack of attention from my parents or teachers. I realize if I were a more naturally
modest, decent fellow, I would have insisted on my original name of "Accelerated
Pipcount" or even "Speed Counting".
But it is too late to pretend.
Anyway, take your pick.
------------------------------------------------------------
Other
Pipcount Methods
The most
common way to count total pips is by the "Straight Count". The number of checkers on each point are
counted, weighted by their point numbers from 1 to 24. If there are two checkers on the "23"
point, they count as 46. Three
checkers on the "14" point count as 42, and so on. These weighted subtotals are added
together to determine the total number of pips necessary to bear them all
off. To function as a human
calculator can be tedious, though the process does get easier as one gains
familiarity with multiples of the point numbers which arise most
often.
Jack
Kissane's "Cluster Counting" improves on straight counting by isolating commonly
found checker clusters having pipcount multiples of 10 (or 5). Mental shifting can produce these
clusters (other checkers to be moved the opposite way to compensate), or pips
left over are added or subtracted.
Basically, Cluster Counting utilizes several helpful reference clusters
with straight counting as a fallback.
There are
two other interesting pipcount methods which I heard about only after
near-completion of this article:
One is Mark
Denihan's "Quadrant Crossover" technique (outlined in an article by Mark
Driver). Its basic idea is
startlingly similar to one of my prototypes for Naccel: Count the quadrants, multiply by 6; to
count the remaining pips visualize the 1, 2, 3, 4, 5 and 6th points stacked on
top of each other, and add the six numbers together. Obviously, since I did not stick to that
blueprint myself, I believe Naccel to be a substantial
improvement.
The other is
Douglas Zare's half-crossover method, which weights 8 half-quadrants
("triples"), and then adds 75 to get an excellent approximate count. This is a clever idea, as one would
expect from Zare, and easier than straight counting. However, to get an exact count,
Zare says it all himself, when he humorously preludes his article with an
excerpt from Lewis Carroll's classic:
"And you do
Addition?" the White Queen asked.
"What's one
and one and one and one and one and one and one and one and one and
one?"
"I don't
know," said Alice. "I lost
count".
------------------------------------------------------------
How
does Naccel work?
The basic
precept for Naccel is that each checker is required to travel through a specific
number of QUADRANTS to bear IN, and through a specific number of POINTS to bear
them OFF. One need not identify,
let alone multiply, the "19" or "21" point or such -- there is nothing higher
than the 6 point!
The opening
position for Black's checkers is illustrated in Diagram A below. Black's Quadrants are marked "0", "1",
"2", and "3" and "4" for a checker on the roof. In the text, we will refer to them as
"Q0" thru "Q4". Just as in the
counting of points, each unit is a "Pip"; in the counting of Quadrants, each
unit is a "Quad".
Notice that
the Quadrant divisions are shifted one point. This will be easy to get used to, I
promise, and the reason for it is simple:
It is more efficient to count 1:0 ("6-point") checkers as a Quad in and
of themselves (or 2, 3 or 4 quads, in the case of the "12", "18" or "24"
points), rather than as 0:6 (or 1:6, 2:6, or 3:6), which would mean being
saddled with six residual pips for each one.
Checkers are
pip-defined by the point numbers (0 thru 5) on which they stand -- the number of
pips required to bear INTO the next quadrant. Look at the number AFTER the
colon of a notated point; for example, "1:2" has two residual
pips.
It is
difficult to emphasize strongly enough that becoming accustomed to calling the points
by their new names will help you enormously in using Naccel. For purposes of this article, we will
refer to points by their new (mod-6) notation, though we will usually list the
traditional point numbers in parentheses as a reference.
For example,
what is classically known as the "10 point", is Quadrant 1, Point
4, so we designate it "1:4".
The "23 point" is Quadrant 3, Point 5, or
"3:5". [Note that
multiplying the number before the
colon by 6 and adding the number after the colon translates back to the
traditional number of any point].
Checkers in
Q0 will mostly be referred to by their classical name; thus 0:5 can still be
called the 5 point. The "13" point
is now "2:1", and the "7" point "1:1", though we will mostly refer to these by
their descriptive names, the "midpoint" and "bar point".
Diagram A
Okay -- How
do we count, using these new-fangled point numbers?
------------------------------------------------------------
How
to do a TOTAL Pipcount
·
Step 1: Count the number of
"Quads".
·
Step 2: Add to that the
"Squads".
·
Step 3: Count any leftover
pips.
------------------------------------------------------------
We will now
clarify these steps, as we count Black's starting position
(diagram):
(1) Count the checkers in each
quadrant, as if they are on four "big points", in order to determine the number
of Quads necessary to BEAR IN all our
checkers:
The 2 back checkers ("24pt") in "Q4" count 4. (2 x 4) = 8.
The 5 checkers (on midpoint) in "Q2" count 2. (5 x 2) = 10.
The 8 checkers (6 and "8" pts) in "Q1" count 1. (8 x 1) = 8.
[8 + 10 + 8] = 26.
(2) The three checkers on the 1:2
("8pt)" are a "Squad" (a 6-pip unit). 26 + 1, that makes 27
altogether.
(3) Count the leftover pips: The five midpoint checkers each count 1
pip. 5 x 1 = 5.
----------------------
I have
outlined the count in great detail above, but if you had been able to listen to
me count under my breath, what you actually would have heard, in a span of three
or four seconds, would have been "8..18..26..27, and 5".
An Attentive
Reader: But where is the rest of
it??
Nack: "There is no more. My count is 27:5. What is yours?
Reader: "Two on the 24 is 48, plus 13
times...65... a hundred and uh.. 13, plus 24.. 137.. and 30.. 167."
Nack: "Perhaps familiarity of the opening
points helped, and there were only four of them, but still... 14 seconds... you
are pretty fast for a straight counter.
And you got the right answer.
But did I notice you splatter a few beads of sweat as you jittered out
the arithmetic?"
Reader: (hiding
smirk well): "Well, at least someone has the total now... What
good does this 27:5 do
you?
Nack: "Technically, it means 27 Quads + 5
pips. I am confident that if I
multiply the 27 by 6, and add the 5, I will arrive at the same total you
got. But I never have the need, as
you will see".
------------------------------------------------------------
Counting
QUADS
This stage
is quite simple. Imagine a
SPEEDBOARD. That is the equivalent
of what you are counting. Your
quadrants are just big points, counted as 4, 3, 2 and
1.
The roof is
your "4 point". The opponent's home
board, your "3 point". The
opponent's outer table, your "2 point".
Your outer table, your "1 point".
Your inner board (below the 6-point) doesn't count at all. So, overall, you have the equivalent of
a speedboard with fewer, sometimes far fewer, than 15
checkers!
I admit, at
least at the outset, that counting will not be as simple as this analogy
suggests. Because the quadrants are
shifted one pip, the bar (and board edges) visually hinder the "6", "24" (and
the less seen "18" and "12"), point checkers, from being assigned to the proper
"points" of your speedboard.
I can assure
you, however, that with a basic understanding of the system, and practice, any
such confusion will rapidly disappear.
You will find ways to remember; for example, it is easy to toss the "6"
point in with the rest of the Q1 checkers when it is its usual conspicuous,
towering self.
Perhaps the
other imperfection in the analogy is that checkers spread out over an entire
board are not side by side, as they are in a speedboard, so your eyes have to
travel a bit further to count. What
you pay for in distance, though, you are sometimes more than refunded in
compactness. For example, it is
faster for me to count an outer 4-prime as 8 checkers, than it is to correctly
count 8 checkers on the 1-point.
There is not a big difference either way -- the ability to count the raw
number of checkers in a quadrant at a glance is, as you might expect, just a
matter of practice.
In learning
to count your Quads, it makes sense to start on one end and sweep
systematically; in this way you can be sure to correctly account for every
checker. I recommend starting with
Q4. Just as you would with a
speedboard, count your big "points" (quadrants) first.
As you gain
confidence, you can improve your speed by selectively combining checkers in
different quadrants in the same way you might combine different points of a
speedboard. For example, you know
that two checkers each on the 3 and 2 points create a "block" of 10 pips. In the same way, two checkers each in Q3
and Q2 create a 3322 "block" of 10 quads. (Hint: The midpoint checkers are the most
common Q2's, by far).
One
Q-pattern that frequently arises is a checker on the 4-0 ("24") point (or
roof), plus an anchor (or two checkers split in the opponent's home
board). This 433 pattern
also counts as 10 quads. Or, the
anchor can instead be three on the midpoint, say, to create another useful
10-quad pattern of 4222.
Five
checkers in Q2 (e.g., on the mid), or 22222, will become as obvious a
10-quad pattern as five checkers on the 2-point is a 10-pip pattern. Eventually, you can introduce other
patterns, such as 222211, or 2221111 (usually combinations of
midpoint and "8" point checkers), which occur to you, and the amortized gain
will save more time than the initial cost in finding or remembering them at the
board.
If 10-quad
patterns don't fall into your lap, don't fret. Positions generally range from 12 to 25
total quads, so your options of extracting a convenient block may be
limited. Just retain that image of
a speedboard in your mind. You
might be drudging along, counting quadrant by quadrant, until it occurs to you,
say, that four on the "2-point" plus four on the "1-point" are like having four
on the 1+2 "point". Similarly, four
Q2's plus four Q1's count as 12.
Whatever the pattern is, once you discover it and decide you like it, you
can add it to your repertoire.
Soon, we
will describe the other half of the equation. Then, we will put the two halves
together, and practice full board counts.
------------------------------------------------------------
Origin
of a New Term
With Quads
quickly counted, what is left? The
answer is: All the pips which
checkers require to bear INTO the next quadrant.
My first
system was to scrape these residual pips together into patterns of 10, but
eventually it dawned on me that I could allow the geometry of the backgammon
board to work in Naccel's favor for this phase of the counting as well. I have found 6-pip checker groups
to be prettier, more manageable, and fit much more naturally onto the board than
the 10-pip groups I have subsequently discarded. Most importantly, the use of 6-pip
groups has meant that EVERYTHING can be converted to the equivalent of Quads,
with a mere handful (literally, 0-5) of leftover pips.
I wanted to
think of a catchy name for these 6-pip checker patterns. "Clusters", which is a more aesthetic
name than "Clumps" or "Combos", was already taken, and "Groups" or "Sets" seemed
mundane. "Virtual Quadrants" was
descriptive and catchy, but a bit long, and "Virtuals" sounded
funny.
The next try
was to find names that suggested the number six. Looking for something jazzier than
"Hexads", I got sidetracked to mental depths I barely dare to repeat. The trouble began when "Sex", the
Swedish word for six, was suggested to me, and "Sexes" became the prime
candidate. Individual names for
"Sex" shapes filled my brain:
Checkers in a block shape became a "box", checkers in a stack a "rod",
checkers along the edge a "lay", three on a point a "three-way", and so on. I even had an animated argument with
myself about whether a certain shape looked more like a "spoon" or a
"sucker".
It is a
matter of definition whether I ever recovered my sanity, but it was when the
word "Squad" suddenly struck me, that I was jolted out of my Scandinavian
fantasy. A term that suggests
tactical deployment upon a battlefield, "Squad" gives life to the checkers,
"men"; yet it literally means a group of "people" -- hey, that might even please
the feminists (if such a thing is possible). Also, quadrant rhymes with squadron
(close enough), and Quad with Squad.
Clearly these blood brothers are meant to be a tag-team; it seems only
natural for "Squads" to take over where "Quads" leaves
off.
------------------------------------------------------------
The
Ten Basic SQUAD Patterns
The use of
"Squads" is nothing more than a system by which to quickly and conveniently
count the Residual Pips. This would include each of the checkers on
the 6, 1:6, 2:6 and 3:6 points, except that we had the foresight to shift our
quadrant boundaries (one point), thus including them in the Quadcount. We need only Squadrify the remaining
checkers, pip-defined by the point 1, 2, 3, 4 or 5 on which they
stand.
A "Squad"
is, basically, any six pips.
The Ten basic Squad patterns are incorporated into Diagrams B1 and B2
below (with a few irrelevant checkers on the 1:0 ("6") points so that you can
get used to ignoring them):
Diagram B-1
------------------------------------------------------------
Diagram B-2
<>
Exactly as
in the traditional notation system, everything is reversed from Red's point of
view. To get the Red notation for a
point (until it becomes second nature), you read the point on the exact FAR SIDE
of the board (just as you must do with all classical
diagrams).
Translated
into shorthand, the Ten Basic Squads are:
33, 42, 51, 222, 321, 411, 2211, 3111, 21111, and 111111. A name and description of each
follows:
·
(a) 33. The "PAIR". Starting with the most obvious pattern,
two checkers on the third point of any quadrant add up to 6 pips. This appears in the top diagram as
Black's 3:3 ("21") point, and in the bottom diagram as Red's 3
point.
·
(b) 42. The "SPLIT" can be thought of as
(4+2) = 6, or as two checkers which can be shifted, one a pip forward and the
other a pip backward, to form the 3 point.
In the top diagram, Black has a 42 in his home board. In the bottom diagram, Red's back
checkers form a 42.
·
(c) 51. The "WIDE" (as in wide
split). See top diagram, Red's
bearoff. Again, a mental shift will
transform these two blots into her 3 point.
·
(d) 222. The "DUCK", appears as 3 checkers
on Red's 1:2 ("8") point in both diagrams.
Although Q1 is the most common place to find it, 222 can appear in any
quadrant. A "duck" is a short stack
of stones marking a cross-country trail.
In backgammon, "ducks" refer to deuces.
·
(e) 321. The "LAYER": See Bottom diagram, Black's Q1. It is easy to see this is the bottom
layer of a 321 prime (the "Double Layer").
Whenever we add a layer, we add a unit of Squad. Other Squad patterns can be easily
visualized in combination with (on top of) the Layer.
·
(f) 2211. The "BLOCK". See Top diagram, Q1, or bottom diagram,
Q0, both Black. This is a really
handy formation. If doubled in
height, it becomes the "Building" (or if tripled, the
"Skyscraper").
·
(g) 411. The "WEDGE". Bottom diagram, Black's Q2. As with all basic patterns, these can be
combined from separate quadrants; e.g. 3:4 ("22pt") + two on the
mid.
·
(h) 3111. The "TRIANGLE": Top diagram, Black's Q2. A good resource for a 3-point checker
which can't form a pair, or find a layer.
·
(i) 21111. The "SOCK": Top diagram, Red's Q2. Useful at the end, to round up
1-pointers (midpoint, bar pt, ace pt, roof). See also the "Stack".
·
(j) 111111. The "STACK": Bottom diagram, Red's Q2. Six checkers on, or symmetrical around
any point, convert to an exact number of Squads (in this case, 1). For example, six checkers on a 5 point
count 5 Squads.
------------------------------------------------------------
Multiple
Squad Patterns
The most
useful Multiple Squads are "isolated" -- those which cannot be constructed by
piecing together Singles. That
these Multiples are the Singles' mirror images around the 3 point helps
to reinforce the patterns. (The
Pair, Split and Wide above, and the Kicks below, are their own mirror
images). Multiples arise less often
than Singles, but are worth their weight in gold because they combine the
otherwise clumsy checkers on the 5 and 4 points.
The isolated
Doubles are 552, 543, 444, 5322 and 4431.
They are illustrated, along with the two Triples of 5553 and 5544, and
the one Quadruple of 55554, below in Diagram B-3.
The
Quintuple-Squad 555555 ("Big Stack") is the mirror image of one of our basic
Squad patterns (the 111111 "Stack"), but is so rare, that I have not depicted
it.
Diagram
B-3
·
(a) 552. The "FIN", or "Big Wedge" is in
Black Q1. A "fin" is jargon for a
five-dollar bill, and this pattern also resembles a shark's fin. A theme in all multiple-5 formations,
The "2" gives one pip to each of the 5's and tips you off that the formation
counts 2 Squads.
·
(b) 543. The "BIG LAYER" (2 Squads), is
the mirror image of the 321 Layer (and the same ideas for building up
apply). If you look at Black's home
board, you will see a Big Layer partially lurking under other checkers (as
layers often do). Remove it, then
take away a "Fin", and you will see that the home board is left only with a
"Wide" (51), which comes to a total of Five Squads.
·
(c) 444. The "FORCE" ("May the fours be with you is a popular
backgammon pun"), or "Big Duck" (2 Squads) is in Red Q0. It is easy to combine checkers from
different quadrants, e.g. a 1:4 ("10-point") blot hops forward exactly one
quadrant, or over from the 3:4, to stack onto the four
point.
·
(d) 4431 and 5322. The "KICK" (looks like a foot
kicking a soccer ball) -- see Red's checkers in both outer boards. This is the only pattern whose mirror
counts the same (2 Squads), hence only one name. An easy shift (one checker back, the
other forth, one pip) transforms either Kick into the 4422 "Double
Split".
·
(e) 5544. The "BIG BLOCK" (3 Squads) is in
Black's Q3. The mirror image of the
"Block" (2211), 5544 is great for ridding the least combinable checkers. If twice as high, it is called the "Big
Building".
·
(f) 5553. The "BIG TRIANGLE" (3 Squads; the
"3" part tells you so). In Black's
home board, this can be removed to leave 5421, which I call the "Split Layer" (a
combination of the 42 Split and the 51 Wide); it adds 2 more Squads, for a total
of 5 Squads in the home board.
·
(g) 55554. The "BIG SOCK" (4 Squads, as the
"4" indicates). Of all the ways we
have counted this particular Black home board, using the Big Sock is
easiest. Left over is simply a
Layer (which again brings the count to 5 Squads).
"One
even, two odd" is a useful rule to apply to all formations, including the
Squad Combinations below: When
there is one blot (or a lone checker on top of a primish squad), it is
always on an even-numbered point.
If there are two blots, they will both be on odd
points.
------------------------------------------------------------
Squad
Combinations
Most
multiple Squad patterns are combinations of two or more Single ("basic")
patterns. Consider these advanced;
knowledge of them is a lower priority than of those illustrated in the "B"
diagrams.
It is great
practice to visualize (or set up) the following combinations, and determine
which single (or multiple) patterns they combine. (Note how the mirror images help to
reinforce the patterns). If you are
not in the mood to do so now, feel free to skip this section and come back to it
later, or use it as a reference.
·
2
Squads: 4332 = "Hat". 54111 = "Wide Wedge." 441111 = "Double Wedge". 44211 = "Drop-kick". 43311 = "Tandem". 4422 = "Double Split". 5511 = "Double Wide". 5421 = "Split-Layer". 332211 = "Double Layer'. 33222 = "Chair". 3222111 = "Boot". 322221 = "Top Hat". 333111 = "Odds". 22221111 =
"Building".
·
3
Squads: 333222111 = "Triple Layer". 433332 = "Big Top Hat". 44433 = "Big Chair". 4433211 = "Truck". 443322 = "Tri-pair". 444222 = "Triple Split". 555111 = "Triple Wide". 55521 = "Wide Fin". 553311 = "Short Odds". 55332 = "Big Tandem". 55422 = "Big Drop-kick". 554211 = "Feet-In". 544221 = "Feet-Out". 543321 =
"Sombrero".
·
4
Squads: 555333 = "Big Odds". 555531 = "Wide Triangle". 544443 = "Giant Top Hat". 5543322 = "Big Truck". 55433211 = "W". 444332211 = "Fourplex". 554433 = "Double Big Layer". 55442211 = "Split Blocks" or "Double
Split-Layer". [8-4] Prime =
"Special 5-Prime".
·
5
Squads: 5554443 = "Big Boot". 554433222 = "Big Fourplex". The [1-5] Prime or any "Six-Prime" also
counts 5 Squads.
------------------------------------------------------------
Squadcount
Trouble-Shooting Guide
This is the
final preparation for counting full boards (and also a reference for
later). A comprehensive look at the
pitfalls which are possible during a Squadcount will prevent you from being led
astray:
·
(1) Far
Side confusion: This is the
most likely cause of a miscount.
There can be a tendency, at first, to confuse the middle points in Q2
and, especially, in Q3. This
is because Red's 2 and 3 points are Black's 5 and 4 points, and vice versa.
Remedy: Until instant recognition sets in, just
keep reminding yourself which direction the checkers are traveling for the side
you are counting.
·
(2)
Shrinkage: Uncounted
checkers disappear as if they were counted.
·
(3)
Ghosting: Checkers you have
taken off the board reappear, causing you to combine/count them again.
Remedy for
both: Sweep systematically, as if you are
vacuuming a rug, so that you are less likely to forget or redo a corner. Avoid darting around, leaving isolated
checkers or groups. If you collect
only part of a point, "vacuum" the rest of it as soon as
possible.
·
(4)
Pattern Confusion: 2111 is
too short to be a sock. 432 is a
bogus layer. 441 is not a real
wedge. 5422 is a footless
kick. 3322 or 4433 is a phony
block.
Remedy: Verify that patterns total 6, or a
multiple of 6. Review Diagrams B1
thru B3, and keep practicing your counts.
Getting these patterns right is mainly a matter of repetition; soon, the
misfits will just plain look wrong.
·
(5)
OverSquads: With proper
combining, these rarely arise, but, occasionally, your final remainder will be
more than 6 pips, with no squads available. These rogue combinations are 322, 431,
441, 443, 522, 532, 553, 544, 5554, 55555, and certain
subsets.
Remedy: Review the Multiple Squads section, and
focus on combining 5's and 4's first.
Ration low point-count checkers (e.g., midpoint and 1:2). If you do oversquad, just shift the high
checker(s) up to the 6 point, to create a 1-checker squad, or shift to make the
3 point. For example, 553 or 544
can shift to 661 = 2:1, or 443 becomes 533 = 1:5. This is a one-time deal, and occurs
last; thus, it is easy to add to your (s)quad total.
·
(6) Crossover Shift: What appears to be an innocent shift
between the "6 point" and a lower point, actually crosses over a quadrant
boundary. (By contrast, note that
shifting 1:0 to 1:1 ("6" to "7"), or vice versa, is fine, because that does not
cross over). Similarly, with the
"24", "18" or "12" points, though the temptation to shift there is relatively
infrequent.
Remedy: Make your shift BEFORE counting the
original Quads (and then so as not to forget, start the Quad count from that
end). If you do shift after,
subtract 1 Quad if shifting 6-5 (or add a Quad if 5-6). Or, sometimes you can plug the desired
prime holes by "Quad-hopping" (shifting checkers 6 pips), which can be
done freely, without burdensome side effects. The text accompanying Diagram D contains
illustrations of this theme.
------------------------------------------------------------
QUADS
and SQUADS -- Putting it All Together
Yippee! It is time to apply the techniques we
have learned to count full boards.
Take another
look at Diagrams B1, B2 and B3.
Count the Quads, and then add the Squads, as you go, to the Quads
(sub)total you got. Do not convert
to pips. Write down your Quad
totals for Black, for Red, and the Quad difference. Then compare them with the answers,
below.
[Please note
that the term "Quads" refers both to the quadrant units first counted, and also
to the combined total of these quadrant units and the Squads added to them. In context, it is usually easy to see
which one is being referred to, but if there is possible confusion, then the
terms "Original", or conversely "Combined" (or "Total") can
preface].
If you feel
slow, or lose track of the Naccel procedure, or get the wrong answers, review
"How to do a Total Pipcount", "Counting Quads", or the Squad sections, as
needed, and try again.
----------------------
Diagram
B-1 --
Black:
Quads: The anchor (which is two checkers in Q3)
counts as 6 quads. The clump of 4
checkers in Q1 is convenient to add to it, because that makes 10. The three on the 1:0 ("6-pt") are also
Q1 checkers (remember), so that's 3 more quads, making 13 so far. The four checkers in Q2 count 8, for a
total of 21
Quads.
Squads: Starting from his back checkers and
sweeping around: Black has a Pair,
a Triangle, a Block, and a Split, for a total of 4 Squads. Adding that to the 21 original Quads,
makes 25 Quads
altogether.
Diagram
B-1 --
Red:
Quads: The five in Q2 are nice, that makes 10
Quads. There are 8 more in Q1, for
a total of 18
Quads.
Squads: Starting from the midpoint area: Red has a Sock, a Duck, and a Wide, for
a total of 3 Squads. That makes 21 Quads
altogether.
Summary: Red leads 21 Quads to 25, a difference
of 4
Quads.
Diagram
B-2 --
Black:
Quads: Black has three in Q2, which count as 6
quads, plus 8 more in Q1, makes a total of 14 Quads.
Squads: Black has a Wedge, a Layer, and a Block;
that's 3 Squads. That makes 17 Quads
altogether.
Diagram
B-2 --
Red:
Quads: Red's back checkers count 6, the
midpoint is 12, plus 5 in Q1, makes 23 Quads.
Squads: Red has a Split, a Stack, a Duck, and a
Pair; that's 4 Squads. That makes 27 Quads
altogether.
Summary: Black leads 17 to 27, a difference
of 10
Quads.
Diagram
B-3 --
Black:
Quads: Black's back checkers (four in Q3) count
12, plus 3 in Q1, makes 15
Quads.
Squads: Black has a Big Block (counts 3), a Fin
(2), a Big Sock (4), and a Layer (1), for a total of 10 Squads. Added to the 15 Quads, that's 25
altogether.
Diagram
B-3 --
Red:
Quads: Four Q2 checkers make 8, plus 8 in Q1,
makes 16
Quads.
Squads: Red has two Kicks (or two Double Splits
by virtue of shifting) -- each counting 2, so that's 4 so far. The Force in the inner board counts as
another 2, for a total of 6
Squads. Added to the 16, that's 22 Quads
altogether.
Summary: Red leads 22 to 25, a difference of 3 Quads.
------------------------------------------------------------
Counting
a Real Game Position
Squads are
usually not ALL as conveniently arranged as in the B diagrams. Let us try counting a position which
arose in an actual game -- a tricky middle-game example from New Ideas in
Backgammon (Woolsey/Heinrich) #21 (p. 60), illustrated
below.
Admittedly,
we would not count this position in live play. No matter what we discover the race to
be, it is too risky to break the anchor and hit with the 6-3 rolled in the
actual game. Nor can either side
consider a double based on the race until there is a shot, serious
deterioration, or some sort of contact is broken. We are counting this position purely for
practice.
Starting
with this diagram, I recommend you pull out the original books (if you have
them), and open to the page from which I've borrowed the diagrammed positions
for this article (or print or photocopy from here). This will avoid having to scroll your
screen or turn pages back and forth in an attempt to follow
explanations.
One final
recommendation, before beginning:
To best benefit from this article, make the effort to understand each
adjustment in each step of the full counts offered under each diagram, before
moving on. At first this may seem
tedious, but when you catch on, your reward is that your mind will probably feel
a bit like a rocketship at takeoff.
Again, count
on your own (Quads + Squads, and leftover Pips, please). Write down your steps, and compare your
answers to those immediately below the diagram:
Diagram C
Diagram
C --
Black:
·
Quads: Two Q3's is 6, plus 12 (in Q1) =
18.
·
Squads: The 3:3 point is a "Pair", for 19. The 5544 is a "Big Block" -- that's
22. The "Duck" makes 23, with just 1 leftover pip (ace
point).
Diagram
C --
Red:
·
Quads: Two in Q3 and two in Q2 (the 3322 combo)
make 10, plus 7 in Q1 makes 17.
·
Squads: The 3 point "Pair" -- that's 18. The 5 point plus a 1:2 checker create a
"Fin" (counts 2) -- that's 20 -- which leaves a "Block" outside, that's 21. The 2:4 goes with one of the 3:2 for a
"Split", makes 22, with 3 leftover pips (3:2 +
mid).
TOTALS: So, Black has 23:1, and Red has 22:3. Red leads by 4 pips. [If you don't yet see this difference
easily, count on your fingers up from 22:3, thusly: "22:4, 22:5, 23:0,
23:1"].
----------------------
How did you
do? Your count was wrong? Okay: Try to figure out how it happened. And don't worry. There's a significant chance that you
will make a mistake or two the first few times.
I gave a
longer solution in the text above, but actually I lucked out and counted Black's
entire position in under 3 seconds.
"6..18... wow -- 23, and 1."
Can you guess what I had done?
I had noticed that Quad-hopping the 3:3 point around and inserting it
into the 1:3 slot formed the "Big Fourplex" pattern (worth 5
Squads).
Before long,
squads will jump out at you right and left, even when combined from different
quadrants (believe it or not), and you will be counting them as confidently as
chairs around a table. There will
be many choices, but finding squads which use up the 5 and 4 point checkers
first will retain more flexibility for coralling the rest of the checkers. You can let that principle guide your
sweep, as it did Red in the above position.
Red went
straight for the 5 point (knocking off the Pair on her way, so as not to leave
it isolated), and had a choice of 2-point checkers to combine. She chose the one outside because it
left a Block there, but she could just as easily have grabbed one of the 3:2
checkers, combined the remaining one with the 2:4 (a Split), and shot down the
Duck, with 3 one-pointers left over (bar point and midpoint).
An
experienced counter would likely see Red's 3:2 point as 4 pips to be combined
onto the 2:4 blot, and quad-hop them to the 4-point for a "Six-prime" (5
Squads), with 3 pips left over.
------------------------------------------------------------
Counting
the RACE
We utilized
the last diagram mainly to practice a full count. In actual play, the only good reason for
needing to know the TOTAL count for both sides (as opposed to just a Comparison
count) is in order to decide whether to double, to redouble, or to take or pass
a cube in a straight race (or light contact
position).
Consider the
position below. Should Black
double? Redouble? Should Red take?
Let's start
by determining the total count (Quad + Pip format) for both sides. Do NOT convert the Totals to
Pips, only the difference to Pips.
Don't worry: In the next
section, I will show you how to make accurate cube decisions with the same
Naccel-style numbers we have been producing, and with greater ease than you have
ever experienced with straight pipcounts!
Diagram D
Diagram
D --
Black:
·
Quads: Two Q2's (midpoint) is 4, plus 10 (in
Q1) = 14.
·
Squads: The 4 point + 1:2 ("8") point (a great
"Double-Split" to know), makes 16, plus a "Layer" makes 17. The 5 and 1:5 team up with the midpoint
for two Wides, making 19. There are 0 leftover
pips.
Diagram
D --
Red:
·
Quads: Four Q2's is 8, plus 8 (in Q1) =
16.
·
Squads: 33222 is the "Chair" (Pair + Duck), for
18. The 5 point combines with two
1-pointers on the mid for two Wides, that's 20. There are 5 pips left over (the 3 pt blot
+ two mids).
TOTALS: Black leads 19:0 to 20:5, for a difference of 1:5,
or 11
pips.
----------------------
Before we
determine cube decisions, note that this diabolical diagram contains a couple of
advanced pitfalls. Knowledge of the
Special 5-Prime (8 thru 4 pts), could be just enough to get you in trouble. If you shift the 6-point checkers, one
forward and the other backward, you have achieved that
prime.
However, under the Squadcount Trouble-Shooting section, a warning is issued (with explanation). Shifting from 6