a[Naccel]erated Pipcount
byNack Ballard
Count-- On Me?
"Twelve plusfifteen plus six in my board, that's thirty-three, plus sixteen for thetwo on myeight point, that's forty-nine, plus nine equals fifty-eight, and fourteenequals 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'treally paying attention... sixty-six?" my doubles partner suggesteddubiously. "You had just countedthe 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 wassupposed to add it TO..." Here I goagain, I thought... now I'll have to start all over again for the tie-breakingcount. "What were my first twotries?"
"Hmm....theydiffered from each other by eleven... I remember that..." his voice faded asquickly as his grin.
I couldalready feel my equity shrinking. "Well, let's start again with what we know. What did we get for the count on theirside, 130-what?"
"I know itwas my JOB to remember the counts", he savored the word like a sour plum. "But when you asked me about your lastpipcount, it all went out of my head."
"I wasn'tasking you, I was asking myself". Hmm, that didn't come out right.
"Well, myability to distinguish rhetoric isn't what it used to be, nor is my memory. I can't even recall why I paid bothhalves of our entry fee."
I was nolonger sure myself of the reason. Iwanted to quip back on his misuse of the word "rhetoric", but someone had topull the team back together.
"Okay,LOOK..." Pretending I was still incharge 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 withTHAT."
... Soundfamiliar? Well, maybe your presenceof mind is not quite as absent as mine, but might not such a calamity occur ifyou find yourself deprived of enough sleep, or otherwise lackingfocus?
Countingpips costs time, and drains our reserve of energy. It is important to be able to stayintent on the positional considerations, which, if assessed correctly, will morelikely guide us to correct plays and cube decisions.
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Followinga Dream
With thegood of the human race in mind, then-- well okay, I admit it was because I wearied of my own incompetence -- Iactually 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 poppeda cyberboard with the four quadrants taking turns bulging out atme.
The nextmorning, while driving, I mused how easy counting the race would be if all I hadto do was count FOUR BIG points instead of 24 little ones. No more two-digit numbers to add ormultiply.
Thinkingfurther, it seemed as if this simple scheme would indeed give a goodapproximation. Beginning playerscould benefit by being taught to initially adopt a Quadrant Count, to geta feel for how much they were ahead or behind.
Then it hitme. An expert player could use thisabsurdly simple counting method, too. The only other step would be to count the remaining pips in eachquadrant!
When Iarrived home, I set up some backgammon positions. To the checkers in my home board Iassigned a quandrant value of 1, to those in my outer tables 2 or 3, and to myback checkers 4. Counting the totalquadrants was as easy as counting a bearoff in which my checkers were all on the4, 3, 2 and 1 points.
To do fullpipcounts, I altered the designations of the quadrants (from 1, 2, 3, 4) to0, 1, 2, and 3, in order to allow residual pips to be ADDED instead ofsubtracted. (A checker on the "22point" needs to travel THREE (not four) quadrants in order to bear IN, and then4 more pips to bear OFF). Hmm...even easier!
I startedleafing through any backgammon literature I could find which had pipcountsaccompanying the diagrams. Atfirst, I made a lot of silly mistakes, overlooking leftover pips. But, as I kept practicing, and graduallystreamlined the method (in particular, after introducing "squads"), I wasastounded to find myself rattling off correct pipcounts in a matter of seconds.
A wave ofeuphoria, bottled up by 25 years of slavish pipcount, washed over me. I shared the details of my discoverywith a friend, Ulf Wostner, who mirrored my enthusiasm by offering to set up awebsite (see the end of this article) to teach this new method of countingpips.
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 imaginethis appellation as some alternate way of spreading my genes. I think it is only fair to clarify thatI attribute neither my psychological imbalance, nor my need to compensate, toany lack of attention from my parents or teachers. I realize if I were a more naturallymodest, decent fellow, I would have insisted on my original name of "AcceleratedPipcount" or even "Speed Counting". But it is too late to pretend. Anyway, take your pick.
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OtherPipcount Methods
The mostcommon way to count total pips is by the "Straight Count". The number of checkers on each point arecounted, weighted by their point numbers from 1 to 24. If there are two checkers on the "23"point, they count as 46. Threecheckers on the "14" point count as 42, and so on. These weighted subtotals are addedtogether to determine the total number of pips necessary to bear them alloff. To function as a humancalculator can be tedious, though the process does get easier as one gainsfamiliarity with multiples of the point numbers which arise mostoften.
JackKissane's "Cluster Counting" improves on straight counting by isolating commonlyfound checker clusters having pipcount multiples of 10 (or 5). Mental shifting can produce theseclusters (other checkers to be moved the opposite way to compensate), or pipsleft over are added or subtracted. Basically, Cluster Counting utilizes several helpful reference clusterswith straight counting as a fallback.
There aretwo other interesting pipcount methods which I heard about only afternear-completion of this article:
One is MarkDenihan's "Quadrant Crossover" technique (outlined in an article by MarkDriver). Its basic idea isstartlingly similar to one of my prototypes for Naccel: Count the quadrants, multiply by 6; tocount the remaining pips visualize the 1, 2, 3, 4, 5 and 6th points stacked ontop of each other, and add the six numbers together. Obviously, since I did not stick to thatblueprint myself, I believe Naccel to be a substantialimprovement.
The other isDouglas 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 wouldexpect 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 anexcerpt from Lewis Carroll's classic:
"And you doAddition?" the White Queen asked.
"What's oneand one and one and one and one and one and one and one and one andone?"
"I don'tknow," said Alice. "I lostcount".
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Howdoes Naccel work?
The basicprecept for Naccel is that each checker is required to travel through a specificnumber of QUADRANTS to bear IN, and through a specific number of POINTS to bearthem OFF. One need not identify,let alone multiply, the "19" or "21" point or such -- there is nothing higherthan the 6 point!
The openingposition 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 thecounting of points, each unit is a "Pip"; in the counting of Quadrants, eachunit is a "Quad".
Notice thatthe Quadrant divisions are shifted one point. This will be easy to get used to, Ipromise, and the reason for it is simple: It is more efficient to count 1:0 ("6-point") checkers as a Quad in andof 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 beingsaddled with six residual pips for each one.
Checkers arepip-defined by the point numbers (0 thru 5) on which they stand -- the number ofpips required to bear INTO the next quadrant. Look at the number AFTER thecolon of a notated point; for example, "1:2" has two residualpips.
It isdifficult to emphasize strongly enough that becoming accustomed to calling the pointsby their new names will help you enormously in using Naccel. For purposes of this article, we willrefer to points by their new (mod-6) notation, though we will usually list thetraditional point numbers in parentheses as a reference.
For example,what is classically known as the "10 point", is Quadrant 1, Point4, so we designate it "1:4". The "23 point" is Quadrant 3, Point 5, or"3:5". [Note thatmultiplying the number before thecolon by 6 and adding the number after the colon translates back to thetraditional number of any point].
Checkers inQ0 will mostly be referred to by their classical name; thus 0:5 can still becalled the 5 point. The "13" pointis now "2:1", and the "7" point "1:1", though we will mostly refer to these bytheir descriptive names, the "midpoint" and "bar point".
Diagram A
Okay -- Howdo we count, using these new-fangled point numbers?
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Howto do a TOTAL Pipcount
� Step 1: Count the number of"Quads".
� Step 2: Add to that the"Squads".
� Step 3: Count any leftoverpips.
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We will nowclarify these steps, as we count Black's starting position(diagram):
(1) Count the checkers in eachquadrant, as if they are on four "big points", in order to determine the numberof Quads necessary to BEAR IN all ourcheckers:
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 27altogether.
(3) Count the leftover pips: The five midpoint checkers each count 1pip. 5 x 1 = 5.
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I haveoutlined the count in great detail above, but if you had been able to listen tome count under my breath, what you actually would have heard, in a span of threeor four seconds, would have been "8..18..26..27, and 5".
An AttentiveReader: But where is the rest ofit??
Nack: "There is no more. My count is 27:5. What is yours?
Reader: "Two on the 24 is 48, plus 13times...65... a hundred and uh.. 13, plus 24.. 137.. and 30.. 167."
Nack: "Perhaps familiarity of the openingpoints helped, and there were only four of them, but still... 14 seconds... youare 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 outthe arithmetic?"
Reader: (hidingsmirk well): "Well, at least someone has the total now... Whatgood does this 27:5 doyou?
Nack: "Technically, it means 27 Quads + 5pips. I am confident that if Imultiply the 27 by 6, and add the 5, I will arrive at the same total yougot. But I never have the need, asyou will see".
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CountingQUADS
This stageis quite simple. Imagine aSPEEDBOARD. That is the equivalentof what you are counting. Yourquadrants are just big points, counted as 4, 3, 2 and1.
The roof isyour "4 point". The opponent's homeboard, your "3 point". Theopponent'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 ofa speedboard with fewer, sometimes far fewer, than 15checkers!
I admit, atleast at the outset, that counting will not be as simple as this analogysuggests. Because the quadrants areshifted one pip, the bar (and board edges) visually hinder the "6", "24" (andthe less seen "18" and "12"), point checkers, from being assigned to the proper"points" of your speedboard.
I can assureyou, however, that with a basic understanding of the system, and practice, anysuch 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 theother imperfection in the analogy is that checkers spread out over an entireboard are not side by side, as they are in a speedboard, so your eyes have totravel a bit further to count. Whatyou pay for in distance, though, you are sometimes more than refunded incompactness. For example, it isfaster for me to count an outer 4-prime as 8 checkers, than it is to correctlycount 8 checkers on the 1-point. There is not a big difference either way -- the ability to count the rawnumber of checkers in a quadrant at a glance is, as you might expect, just amatter of practice.
In learningto count your Quads, it makes sense to start on one end and sweepsystematically; in this way you can be sure to correctly account for everychecker. I recommend starting withQ4. Just as you would with aspeedboard, count your big "points" (quadrants) first.
As you gainconfidence, you can improve your speed by selectively combining checkers indifferent quadrants in the same way you might combine different points of aspeedboard. For example, you knowthat two checkers each on the 3 and 2 points create a "block" of 10 pips. In the same way, two checkers each in Q3and Q2 create a 3322 "block" of 10 quads. (Hint: The midpoint checkers are the mostcommon Q2's, by far).
OneQ-pattern that frequently arises is a checker on the 4-0 ("24") point (orroof), plus an anchor (or two checkers split in the opponent's homeboard). This 433 patternalso counts as 10 quads. Or, theanchor can instead be three on the midpoint, say, to create another useful10-quad pattern of 4222.
Fivecheckers in Q2 (e.g., on the mid), or 22222, will become as obvious a10-quad pattern as five checkers on the 2-point is a 10-pip pattern. Eventually, you can introduce otherpatterns, such as 222211, or 2221111 (usually combinations ofmidpoint and "8" point checkers), which occur to you, and the amortized gainwill save more time than the initial cost in finding or remembering them at theboard.
If 10-quadpatterns don't fall into your lap, don't fret. Positions generally range from 12 to 25total quads, so your options of extracting a convenient block may belimited. Just retain that image ofa speedboard in your mind. Youmight 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 fouron the 1+2 "point". Similarly, fourQ2's plus four Q1's count as 12. Whatever the pattern is, once you discover it and decide you like it, youcan add it to your repertoire.
Soon, wewill describe the other half of the equation. Then, we will put the two halvestogether, and practice full board counts.
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Originof a New Term
With Quadsquickly counted, what is left? Theanswer is: All the pips whichcheckers require to bear INTO the next quadrant.
My firstsystem was to scrape these residual pips together into patterns of 10, buteventually it dawned on me that I could allow the geometry of the backgammonboard to work in Naccel's favor for this phase of the counting as well. I have found 6-pip checker groupsto be prettier, more manageable, and fit much more naturally onto the board thanthe 10-pip groups I have subsequently discarded. Most importantly, the use of 6-pipgroups has meant that EVERYTHING can be converted to the equivalent of Quads,with a mere handful (literally, 0-5) of leftover pips.
I wanted tothink of a catchy name for these 6-pip checker patterns. "Clusters", which is a more aestheticname than "Clumps" or "Combos", was already taken, and "Groups" or "Sets" seemedmundane. "Virtual Quadrants" wasdescriptive and catchy, but a bit long, and "Virtuals" soundedfunny.
The next trywas 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", theSwedish word for six, was suggested to me, and "Sexes" became the primecandidate. 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 withmyself about whether a certain shape looked more like a "spoon" or a"sucker".
It is amatter of definition whether I ever recovered my sanity, but it was when theword "Squad" suddenly struck me, that I was jolted out of my Scandinavianfantasy. A term that suggeststactical deployment upon a battlefield, "Squad" gives life to the checkers,"men"; yet it literally means a group of "people" -- hey, that might even pleasethe 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 onlynatural for "Squads" to take over where "Quads" leavesoff.
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TheTen Basic SQUAD Patterns
The use of"Squads" is nothing more than a system by which to quickly and convenientlycount the Residual Pips. This would include each of the checkers onthe 6, 1:6, 2:6 and 3:6 points, except that we had the foresight to shift ourquadrant boundaries (one point), thus including them in the Quadcount. We need only Squadrify the remainingcheckers, pip-defined by the point 1, 2, 3, 4 or 5 on which theystand.
A "Squad"is, basically, any six pips. The Ten basic Squad patterns are incorporated into Diagrams B1 and B2below (with a few irrelevant checkers on the 1:0 ("6") points so that you canget used to ignoring them):
Diagram B-1
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Diagram B-2
Exactly asin the traditional notation system, everything is reversed from Red's point ofview. To get the Red notation for apoint (until it becomes second nature), you read the point on the exact FAR SIDEof the board (just as you must do with all classicaldiagrams).
Translatedinto shorthand, the Ten Basic Squads are: 33, 42, 51, 222, 321, 411, 2211, 3111, 21111, and 111111. A name and description of eachfollows:
� (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 asBlack's 3:3 ("21") point, and in the bottom diagram as Red's 3point.
� (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 theother 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 backcheckers form a 42.
� (c) 51. The "WIDE" (as in widesplit). See top diagram, Red'sbearoff. Again, a mental shift willtransform these two blots into her 3 point.
� (d) 222. The "DUCK", appears as 3 checkerson Red's 1:2 ("8") point in both diagrams. Although Q1 is the most common place to find it, 222 can appear in anyquadrant. A "duck" is a short stackof 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 bottomlayer of a 321 prime (the "Double Layer"). Whenever we add a layer, we add a unit of Squad. Other Squad patterns can be easilyvisualized 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 reallyhandy formation. If doubled inheight, 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 becombined from separate quadrants; e.g. 3:4 ("22pt") + two on themid.
� (h) 3111. The "TRIANGLE": Top diagram, Black's Q2. A good resource for a 3-point checkerwhich can't form a pair, or find a layer.
� (i) 21111. The "SOCK": Top diagram, Red's Q2. Useful at the end, to round up1-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 aroundany point, convert to an exact number of Squads (in this case, 1). For example, six checkers on a 5 pointcount 5 Squads.
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MultipleSquad Patterns
The mostuseful Multiple Squads are "isolated" -- those which cannot be constructed bypiecing together Singles. Thatthese Multiples are the Singles' mirror images around the 3 point helpsto reinforce the patterns. (ThePair, Split and Wide above, and the Kicks below, are their own mirrorimages). Multiples arise less oftenthan Singles, but are worth their weight in gold because they combine theotherwise clumsy checkers on the 5 and 4 points.
The isolatedDoubles are 552, 543, 444, 5322 and 4431. They are illustrated, along with the two Triples of 5553 and 5544, andthe one Quadruple of 55554, below in Diagram B-3.
TheQuintuple-Squad 555555 ("Big Stack") is the mirror image of one of our basicSquad patterns (the 111111 "Stack"), but is so rare, that I have not depictedit.
DiagramB-3
� (a) 552. The "FIN", or "Big Wedge" is inBlack Q1. A "fin" is jargon for afive-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 formationcounts 2 Squads.
� (b) 543. The "BIG LAYER" (2 Squads), isthe mirror image of the 321 Layer (and the same ideas for building upapply). If you look at Black's homeboard, you will see a Big Layer partially lurking under other checkers (aslayers often do). Remove it, thentake 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 popularbackgammon pun"), or "Big Duck" (2 Squads) is in Red Q0. It is easy to combine checkers fromdifferent quadrants, e.g. a 1:4 ("10-point") blot hops forward exactly onequadrant, or over from the 3:4, to stack onto the fourpoint.
� (d) 4431 and 5322. The "KICK" (looks like a footkicking a soccer ball) -- see Red's checkers in both outer boards. This is the only pattern whose mirrorcounts the same (2 Squads), hence only one name. An easy shift (one checker back, theother forth, one pip) transforms either Kick into the 4422 "DoubleSplit".
� (e) 5544. The "BIG BLOCK" (3 Squads) is inBlack'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 "BigBuilding".
� (f) 5553. The "BIG TRIANGLE" (3 Squads; the"3" part tells you so). In Black'shome board, this can be removed to leave 5421, which I call the "Split Layer" (acombination of the 42 Split and the 51 Wide); it adds 2 more Squads, for a totalof 5 Squads in the home board.
� (g) 55554. The "BIG SOCK" (4 Squads, as the"4" indicates). Of all the ways wehave counted this particular Black home board, using the Big Sock iseasiest. Left over is simply aLayer (which again brings the count to 5 Squads).
"Oneeven, two odd" is a useful rule to apply to all formations, including theSquad Combinations below: Whenthere is one blot (or a lone checker on top of a primish squad), it isalways on an even-numbered point. If there are two blots, they will both be on oddpoints.
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SquadCombinations
Mostmultiple 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 greatpractice to visualize (or set up) the following combinations, and determinewhich single (or multiple) patterns they combine. (Note how the mirror images help toreinforce the patterns). If you arenot in the mood to do so now, feel free to skip this section and come back to itlater, or use it as a reference.
� 2Squads: 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".
� 3Squads: 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".
� 4Squads: 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 "DoubleSplit-Layer". [8-4] Prime ="Special 5-Prime".
� 5Squads: 5554443 = "Big Boot". 554433222 = "Big Fourplex". The [1-5] Prime or any "Six-Prime" alsocounts 5 Squads.
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SquadcountTrouble-Shooting Guide
This is thefinal preparation for counting full boards (and also a reference forlater). A comprehensive look at thepitfalls which are possible during a Squadcount will prevent you from being ledastray:
� (1) FarSide confusion: This is themost likely cause of a miscount. There can be a tendency, at first, to confuse the middle points in Q2and, especially, in Q3. Thisis because Red's 2 and 3 points are Black's 5 and 4 points, and vice versa.
Remedy: Until instant recognition sets in, justkeep reminding yourself which direction the checkers are traveling for the sideyou are counting.
� (2)Shrinkage: Uncountedcheckers disappear as if they were counted.
� (3)Ghosting: Checkers you havetaken off the board reappear, causing you to combine/count them again.
Remedy forboth: Sweep systematically, as if you arevacuuming a rug, so that you are less likely to forget or redo a corner. Avoid darting around, leaving isolatedcheckers or groups. If you collectonly part of a point, "vacuum" the rest of it as soon aspossible.
� (4)Pattern Confusion: 2111 istoo short to be a sock. 432 is abogus layer. 441 is not a realwedge. 5422 is a footlesskick. 3322 or 4433 is a phonyblock.
Remedy: Verify that patterns total 6, or amultiple of 6. Review Diagrams B1thru B3, and keep practicing your counts. Getting these patterns right is mainly a matter of repetition; soon, themisfits will just plain look wrong.
� (5)OverSquads: With propercombining, these rarely arise, but, occasionally, your final remainder will bemore than 6 pips, with no squads available. These rogue combinations are 322, 431,441, 443, 522, 532, 553, 544, 5554, 55555, and certainsubsets.
Remedy: Review the Multiple Squads section, andfocus 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 highchecker(s) up to the 6 point, to create a 1-checker squad, or shift to make the3 point. For example, 553 or 544can shift to 661 = 2:1, or 443 becomes 533 = 1:5. This is a one-time deal, and occurslast; thus, it is easy to add to your (s)quad total.
� (6) Crossover Shift: What appears to be an innocent shiftbetween the "6 point" and a lower point, actually crosses over a quadrantboundary. (By contrast, note thatshifting 1:0 to 1:1 ("6" to "7"), or vice versa, is fine, because that does notcross over). Similarly, with the"24", "18" or "12" points, though the temptation to shift there is relativelyinfrequent.
Remedy: Make your shift BEFORE counting theoriginal Quads (and then so as not to forget, start the Quad count from thatend). 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 desiredprime holes by "Quad-hopping" (shifting checkers 6 pips), which can bedone freely, without burdensome side effects. The text accompanying Diagram D containsillustrations of this theme.
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QUADSand SQUADS -- Putting it All Together
Yippee! It is time to apply the techniques wehave learned to count full boards.
Take anotherlook 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 convertto pips. Write down your Quadtotals for Black, for Red, and the Quad difference. Then compare them with the answers,below.
[Please notethat the term "Quads" refers both to the quadrant units first counted, and alsoto the combined total of these quadrant units and the Squads added to them. In context, it is usually easy to seewhich one is being referred to, but if there is possible confusion, then theterms "Original", or conversely "Combined" (or "Total") canpreface].
If you feelslow, 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, asneeded, and try again.
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DiagramB-1 --Black:
Quads: The anchor (which is two checkers in Q3)counts as 6 quads. The clump of 4checkers in Q1 is convenient to add to it, because that makes 10. The three on the 1:0 ("6-pt") are alsoQ1 checkers (remember), so that's 3 more quads, making 13 so far. The four checkers in Q2 count 8, for atotal of 21Quads.
Squads: Starting from his back checkers andsweeping 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 Quadsaltogether.
DiagramB-1 --Red:
Quads: The five in Q2 are nice, that makes 10Quads. There are 8 more in Q1, fora total of 18Quads.
Squads: Starting from the midpoint area: Red has a Sock, a Duck, and a Wide, fora total of 3 Squads. That makes 21 Quadsaltogether.
Summary: Red leads 21 Quads to 25, a differenceof 4Quads.
DiagramB-2 --Black:
Quads: Black has three in Q2, which count as 6quads, 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 Quadsaltogether.
DiagramB-2 --Red:
Quads: Red's back checkers count 6, themidpoint is 12, plus 5 in Q1, makes 23 Quads.
Squads: Red has a Split, a Stack, a Duck, and aPair; that's 4 Squads. That makes 27 Quadsaltogether.
Summary: Black leads 17 to 27, a differenceof 10Quads.
DiagramB-3 --Black:
Quads: Black's back checkers (four in Q3) count12, plus 3 in Q1, makes 15Quads.
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 25altogether.
DiagramB-3 --Red:
Quads: Four Q2 checkers make 8, plus 8 in Q1,makes 16Quads.
Squads: Red has two Kicks (or two Double Splitsby virtue of shifting) -- each counting 2, so that's 4 so far. The Force in the inner board counts asanother 2, for a total of 6Squads. Added to the 16, that's 22 Quadsaltogether.
Summary: Red leads 22 to 25, a difference of 3 Quads.
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Countinga Real Game Position
Squads areusually not ALL as conveniently arranged as in the B diagrams. Let us try counting a position whicharose in an actual game -- a tricky middle-game example from New Ideas inBackgammon (Woolsey/Heinrich) #21 (p. 60), illustratedbelow.
Admittedly,we would not count this position in live play. No matter what we discover the race tobe, it is too risky to break the anchor and hit with the 6-3 rolled in theactual game. Nor can either sideconsider a double based on the race until there is a shot, seriousdeterioration, or some sort of contact is broken. We are counting this position purely forpractice.
Startingwith this diagram, I recommend you pull out the original books (if you havethem), and open to the page from which I've borrowed the diagrammed positionsfor this article (or print or photocopy from here). This will avoid having to scroll yourscreen or turn pages back and forth in an attempt to followexplanations.
One finalrecommendation, before beginning: To best benefit from this article, make the effort to understand eachadjus