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 eachadjustment in each step of the full counts offered under each diagram, beforemoving on. At first this may seemtedious, but when you catch on, your reward is that your mind will probably feela bit like a rocketship at takeoff.
Again, counton your own (Quads + Squads, and leftover Pips, please). Write down your steps, and compare youranswers to those immediately below the diagram:
Diagram C
DiagramC --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's22. The "Duck" makes 23, with just 1 leftover pip (acepoint).
DiagramC --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 differenceeasily, count on your fingers up from 22:3, thusly: "22:4, 22:5, 23:0,23:1"].
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How did youdo? Your count was wrong? Okay: Try to figure out how it happened. And don't worry. There's a significant chance that youwill make a mistake or two the first few times.
I gave alonger solution in the text above, but actually I lucked out and counted Black'sentire 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 itinto the 1:3 slot formed the "Big Fourplex" pattern (worth 5Squads).
Before long,squads will jump out at you right and left, even when combined from differentquadrants (believe it or not), and you will be counting them as confidently aschairs around a table. There willbe many choices, but finding squads which use up the 5 and 4 point checkersfirst will retain more flexibility for coralling the rest of the checkers. You can let that principle guide yoursweep, as it did Red in the above position.
Red wentstraight for the 5 point (knocking off the Pair on her way, so as not to leaveit isolated), and had a choice of 2-point checkers to combine. She chose the one outside because itleft a Block there, but she could just as easily have grabbed one of the 3:2checkers, combined the remaining one with the 2:4 (a Split), and shot down theDuck, with 3 one-pointers left over (bar point and midpoint).
Anexperienced counter would likely see Red's 3:2 point as 4 pips to be combinedonto the 2:4 blot, and quad-hop them to the 4-point for a "Six-prime" (5Squads), with 3 pips left over.
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Countingthe RACE
We utilizedthe last diagram mainly to practice a full count. In actual play, the only good reason forneeding to know the TOTAL count for both sides (as opposed to just a Comparisoncount) is in order to decide whether to double, to redouble, or to take or passa cube in a straight race (or light contactposition).
Consider theposition below. Should Blackdouble? Redouble? Should Red take?
Let's startby determining the total count (Quad + Pip format) for both sides. Do NOT convert the Totals toPips, only the difference to Pips. Don't worry: In the nextsection, I will show you how to make accurate cube decisions with the sameNaccel-style numbers we have been producing, and with greater ease than you haveever experienced with straight pipcounts!
Diagram D
DiagramD --Black:
� Quads: Two Q2's (midpoint) is 4, plus 10 (inQ1) = 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 midpointfor two Wides, making 19. There are 0 leftoverpips.
DiagramD --Red:
� Quads: Four Q2's is 8, plus 8 (in Q1) =16.
� Squads: 33222 is the "Chair" (Pair + Duck), for18. The 5 point combines with two1-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 11pips.
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Before wedetermine cube decisions, note that this diabolical diagram contains a couple ofadvanced pitfalls. Knowledge of theSpecial 5-Prime (8 thru 4 pts), could be just enough to get you in trouble. If you shift the 6-point checkers, oneforward and the other backward, you have achieved thatprime.
However,under the Squadcount Trouble-Shooting section, a warning is issued (withexplanation). Shifting from 6-5should be done before the original Quad count, or if afterwards, oneneeds to remember to adjust by subtracting one Quad. Deciding never to shift the 6-5 is thesafest policy, though that may deny what could otherwise prove to be a usefulresource.
"Quad-hopping"can often achieve a desired goal without encumbrances. Any time you combine checkers fromdifferent quadrants to form a Squad, you are essentially employing thistechnique. Moving checkers 1 ormore exact Quads maintains their parity, while positioning them intorecognizable patterns. In theposition at hand, Quad-hopping creates the Special 5-prime, worth 4 Squads(cleanly leaving a 5th Squad, the "Layer", left over). This is the fastest way to count Black'sposition.
Even withoutknowledge of the Special 5-prime, hopping both 6's down and covering the 5 pointwith a third clarifies our view. Bybanding the entire army of checkers together, all the ways of combining workableSquads, as well as remainders, are now easier to see: Big Block and a "Boot" (which is Layer +Block); or Fin, Double-Split and a Triangle; or Duck, two Wides and a Kick; andso on.
Red'sposition contains a different trap. Shifting the 1:2 ("8pt) checker forward and the 1:0 ("6pt") checkerbackward (this time it is the okay direction, because it does not step over areal quadrant divider), creates a 5-prime. However, it's the wrong one! Only 5-primes centering around a 3 point [1 thru 5], or sandwiched bythem [8 thru 4], convert to an exact number of Quads.
"Quad-hopping"all four midpoint checkers down to the bar point is useful, though not as a5-prime. We can now more easilycount Red's position as a Double Layer plus either two Wides or a Fin; or aChair and two Wides; either way, with 5 pips left over.
All roadslead to Rome; the preponderance of small counters makes it impossible tooversquad here. A Quad-hop of the3-point checker back to create a Triple Layer still leaves a Wide to grab, witha single 5-pip checker as a remainder. Even if we employ the stinkiest technique possible by using the Sock tomop up all the one-pointers, the [Pair + Fin] or [Pair + Kick] rescues us. How did you do with thisdiagram?
Okay, nowthat we have seen a variety of ways to quickly arrive at the correct totals, wewill figure out what to actually do with our count of 19:0 to 20:5. What are the correct cubeactions?
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NaccelRace Formula
Truncate theLeader's count tojust the Quad number (nothing after the colon). Now,
� If 12 Quadsor more, subtract 1, divide by 2, and round down.
� OR, if lessthan 12 Quads, simply subtract 6.
This is theminimum lead (in pips) necessary for a correct DOUBLE.
Add 1 to getthe minimum redouble, or
Add 4 to getthe maximal take point. [If leaderhas 18+ Quads, add 5].
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That'sit. No percentages. The Naccel Race Formula matchesRobertie's [8 / 9 / 12%] Formula surprisingly well.
Because the[8 / 9 / 12] Formula is not intended for use below 12 Quads (72 pips), Naccel'ssimple [subtract 6] rule then approximates Trice's short-race (no wastage)formula, which is [Subtract 5 from PIPcount, divide by 7, round down for takepoint, subtract from that 3 for redouble or 4 for initialdouble].
For example,the leader has 9:0, 9:1, 9:2, 9:3, 9:4, or 9:5. Truncating, and subtracting 6, Naccelgives a doubling point of 3, a redouble of 4, and a take of 7. Trice's formula agrees with all numbersin this range (54 thru 59 pips). There are a few spots elsewhere in which Naccel's thresholds differ fromTrice's by 1.
It is trickyto find the right balance of close fits and easy implementation, though I amhappy with the way the Naccel formula has turned out. Currently, I am unaware of the existenceof any other reference formulas or tables which would allow me to sharpenRobertie's or Trice's approximations. (Weaver's clever "10% -2 / -1 /+2", though highly practical fortraditional pipcounts, does not offer Naccel accurate enough numbers toemulate).
Please letme know if you find any Naccel examples for which the thresholds stray fromthese or other known formulas, so I can review them for possiblerevision.
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Let's applythe Naccel Race formula to the count of 19:0 to 20:5, difference of 11 pips, we derived from themost recent position:
*** [19 - 1] divided by 2equals 9. ***
So, 9 is our minimum double. 10 is our minimum redouble. 14 is our maximaltake.
(We added 5to derive the take point, because the leader's total is at least 18Quads).
The actualdifference is 11 pips. Black has a solid redouble, and Red hasan easy take.
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Rememberingthe Count
All countingmethods are vulnerable to a common disaster. What if you have completed the count forthe second player, but have forgotten the count of the first player? Well, there's not much you can do atthat point, other than redo the first player's count (and hope you don't then forget the second player's!). This is a case where an ounce ofprevention is better than a pound... on the head.
I heard orread (I don't remember where, or if I'm parroting the method correctly -- sorry)that to remember a traditional count, you could touch a finger of your left handto the outside of the board, immediately in front of the point that correspondsto the first digit (or two) of the count, and a finger of your right hand infront of the point which corresponds to the last digit.
So, 113would be "bookmarked" with your left finger by the 11 point, and your rightfinger by the 3 point. In theory,for 89 you would have your left finger by the 8 point and your right finger onthe 9 point, causing your hands to cross (to avoid confusion with 98), though Isuppose instead holding one finger at an angle in such cases could clarifythat. Presumably, pipcounts higherthan 129 would just ignore the first digit. So, for 130, you could left-finger the 3point and right-finger the bear-off tray.
Possibledrawbacks to this might be that your arms might tire while you are counting theother color, it might inform an observant opponent of the pipcount (in case thatmatters), and it just plain looks silly! However, the method works, and it is easy toremember.
The boardcould be employed in a similar manner to remember Naccel's counts, but because Iwas raised to keep my arms and elbows off the table, I recommend an alternatemethod:
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The"Handy" Count
� Choose aprimary location to rest your left hand (could be on your thigh or alongsideyour chair). Keep track of the lastdigit of your Quadcount by extending 0, 1, 2, 3, 4, or 5 fingers. For a higher digit, move your hand to asecondary location (could be your knee), and extend 1, 2, 3, or 4 fingers todenote 6, 7, 8 or 9 respectively.
� Extend thefingers of your right hand the appropriate number of ResidualPips.
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It isunnecessary to bookmark the first digit of the Quadcount. For example, 12 Quads is virtually abearoff position, and it would be very hard to confuse that with the 22 Quadlength of, say, the opening position.
For bestNaccel results, I recommend use of the Handy Count. As your speed increases, you may wish togradually drop the use of your right hand (residual pips), and eventually evenyour left.
The HandyCount can also be tagged to the Comparison count (see next section). Your left hand can bookmark the Quaddifference (palm up means up Quads, palm down means down Quads),while you figure out the Top-Heavies or Residual Pips.
If you eversee me holding MY fingers up by the board, it will not be a new method. You will know that I'm either having oneof my fits (usually accompanied by slobbering and guttural noises), or I amhoping you'll know how to read the bogus pipcount I will be subliminallysignaling.
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a[Naccel]erated COMPARISON count
� Step 1: Compare Quads.
� Step 2: Compare either Top-Heavies, or ResidualPips.
� Step 3: Add or subtract.
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I recommendthe traditional method of Comparison counting IF, and only if, you findyourself in a position highly symmetrical to your opponent. In this case, sizing up a fewadjustments will be a little faster than Naccel's method. In all other positions, I recommendNaccel.
Forpositions which are difficult to comparison-count using the classical method,Naccel is profoundly practical. Inmany cases, you will be able to stop right after the first step -- the "QuadComparison". While aclose count is not guaranteed, accuracy to the nearest Quad istypical.
What thismeans is that you can often garner sufficient information during Step 1. If you are cubed, and find yourself downseveral quadrants in a light contact position, you are NOT going to take thedouble -- it is merely a waste of time to figure out the relatively small swingin the remaining pips. Top-Heaviesor (especially) Residual Pips are just "fine tuning".
Similarly,if it is a question of doubling, and a simple Quad Comparison reveals you arenot ahead, there is no point in sharpening the count (unless the upper halves ofyour opponent's quadrants look muchheavier than your own). Othercounting methods, which lump it all together, offer no suchrelief.
If you wantto know the race to help you decide whether to make an aggressive or a passivechecker play, a Quad Comparison will nearly always suffice. However, the second step of counting(adjusting either for top-heaviness, or to the precise pip), will always be yourprivilege, should you choose to exercise it.
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ComparingQuads
One approachto comparing Quads is just to count the total Quads for both sides and subtractone from the other, as we have been practicing. However, if we have decided we are notcounting for purposes of applying the race formula (and thus have no need fortotals), we can adopt an even shorter procedure, elaboratedhere.
We do notnormally compare "Speedboards" by counting their pips, but imagine how easy itwould be. You could just go frompoint to point, canceling differences. "I have 2 extra on the 3 point, so I'm down 6 pips. She has one extra on the two point, sonow I'm only down 4 pips. She hasthree extra on the ace point, so altogether I'm only down apip."
ComparingQuads is an identical procedure. While not as vertically compact, the opposite-colored checkers are oneach "big point" are right across from each other, just as in a speedboard, soit is easy to see the differences.
Additionalcancellations can help, as long as one does not have to strain to findthem. If Black has a checker on theroof (Q4), and Red has a couple extra checkers on the midpoint (Q2) instead,these are convenient to cancel. (This is no different, in essence, than Black having a checker on the 4point of a speedboard, while Red has a couple extra checkers on the 2point). Other common cancellationpossibilities include 433 vs 22222 (five on the mid), Or 33 (an anchor) equals 222 (three onthe mid).
Thesesupplemental cancellations can be quite useful in asymmetrical positions. In symmetrical ones, they will tend toincrease the likelihood of a "ghosting" error, and are not needed anyway. Unless a peripheral cancellation is aclear and easy gain, it is more practical to stick to the straightquadrant-by-quadrant comparison.
For purposesof a Quad comparison, it is often convenient to count a 5 or 4 point checker asa Quad. Good examples of this are666 vs 665, or 6666 vs 6664 (or "24 24" vs "24 23"). Pre-canceling such groups is easyvisually, and will likely improve your estimate (4 or 5 pips is closer to 6 thanit is to zero). Even if you plan todo a full Comparison (with residual pips), you can carry around a pip or two forlater adjustment -- you will save not having had to count the Quad one way, andlater the 4 or 5 pips the other way).
If such acancellation does not feel clean or "fair", it may be because you are seeingother 5's and 4's which seem to warrant weight too. That is a signal you should go ahead andperform the standard Quad comparison, and then hone it by applying a morecomprehensive 5-4 adjustment, described in the nextsection.
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ComparingTop-Heavies
Thisoptional supplementary count is done after the pure Quad Count. It quickly estimates the effect of"top-heaviness" by counting the number of 5th and 4th point checkers (in allquadrants) each side has. The ideais to correct the Quad-count, in less time than it takes to perform the totallyaccurate Residual Pipcount.
TheTop-Heavy Adjustment is a trade-off between time and accuracy. It seems warranted when the combinationof (a) and (b) below seems compelling:
� (a) I notice some 5th and/or 4th pointcheckers sitting out there.
� (b) The position is of the type that I don'tneed an exact count, but in case my current count is off a Quad or so, it is somewhat likely that I willmake an error.
Theoretically,one would like to count 5/6 Quads for each excess 5, and 2/3 Quads for eachexcess 4, but that is way too complicated. Even treating 5's differently from 4's seems like more trouble than it isworth. Although you can tailor yourparticular method of adjustment to the level of accuracy you wish to achieve, Irecommend:
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TOP-HEAVYADJUSTMENT: Adjust 1 Quad for each 5 or 4 one colorhas in excess. If the difference is3 or more, Subtract 1 from this adjustment. In the unlikely event the difference is7 or more, subtract 2.
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TheTop-Heavy Adjustment is an easy, level-headed compromise, if you feel the countneeds a bit more accuracy, but not an absolutely pinpoint. [If you then change your mind andfeel, after all, you need the greater refinement of a Residual Pipcount, it islike changing horses in midstream, but still possible: Just back out the T. H.Adjustment].
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CHECKERPLAY based on Quad Comparison
Knowing therace can often help us choose between what I term "Passive vs Aggressive"plays. ("Safe vs Bold", whichconsiders exposing a blot, as in Diagram G, is a subset). For that purpose, I find that tothink of the race difference in terms of a small manageable number of "Quads"easier than some larger number of pips. It allows me to more easily see the big picture. Also, as you will discover, it isusually sufficient to stop after a quick simple Quadcount, withoutrefinement.
Below is anearly bear-in position from New Ideas in Backgammon, # 33 (p.97). Black is considering how to play a rollof 6-4. Is it better to run aroundto the 1:4 point, or to make the outfield anchor (2:4)? Is the race a consideration, and if so,at what race-count would you change your play?
First, writedown your Quad-comparisons and Top-Heavy Adjustments; then check those postedunder the diagram.
Diagram E
DiagramE -- Comparison:
� Quads: Black's back anchor (two Q3's) cancelRed's six outer Q1's. Another BlackQ3 cancels Red's extra 6-point and midpoint checkers. The other two Black Q3's count as 6 Quads.
� Top-Heavies: Black has 4 Top-Heavies, and Red hasonly 2. Adjust Red's lead from 6,to 8Quads.
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Though a 2Quad adjustment is unusual, this Top-Heavy adjustment is a slight over-swing ofthe pendulum (actual count is 7:3, so 7 or 8 was the closest it couldcome).
So, whatabout this Roll of 6-4? Is makingthe outer anchor an aggressive or a passive play? The answer is "passive", even if we grabthe checkers with a great flourish, and jam them into the 2:4 ("16") point,aiming a two-barrelled gun at his midpoint. This will intimidate the opponent onlyuntil next roll, when he starts to realize that what we are actually holding isa water pistol. Running around tothe 1:4 ("10") point is actually the aggressive play, maintaining the backanchor for more real contact. It isimportant to get this conceptually straight, so that we know in which directionthe race deficit will affect our play.
After we geta Quad Comparison of 6, we can ask ourselves: Do we feel that the relative race zonewe have estimated renders our play of 6-4 a close decision? If so, might fine-tuning the countaffect our choice? If no to either,we stop at the Quadcount. If yes,we perform a Top-Heavy Adjustment, or a Residual Pip comparison. (At the board, this "decision" tosharpen the count is all done in an intuitive wink of theeye).
At a racedifference of 8, or 7:3, or even 6 Quads, it is correct to run around (tothe 1:4 point) with the 6-4, and by a huge margin. Kit Woolsey, in his book, offersinsightful analysis, and does particularly well to emphasize how the large racedeficit influences the correct game plan.
By contrast,I sometimes hear the argument put forth, "the race has nothing to do with it --it's only a matter of seeing the timing", but this ignores the fact that raceand timing are closely related. Atsome reduced relative count, various racing or showdown scenarios will become apracticable option; it will become preferable to create a stepping stone, ratherthan to keep forces divided solely on the sole merit of the additional shotequity yielded as a result of clinging to the deepestpoint.
Beinginformed pedantically that a certain move is "clearly" correct, in no wayinvalidates our possible perception that, at the time we had to make a decision,we felt it was beneficial to sharpen our count. What it should do is encourage us tolearn how to better evaluate a particular class of positions, and to mentallyadjust our relationship between certain race parameters and playthresholds.
It turns outin this position that one has to shift the race further than I would haveguessed to swing the correct play. If one moves both ace-point checkers and a 6-point checker back to Red'smidpoint, Black's race deficit has been cut from 7+ to 2+ Quads. Only then does it become correct forBlack to partially abandon the back anchor and make the flexible outside (2:4)point.
For somecube or checker play decisions (though not in this example), you may wish toacquire a count to the exact pip. To this end, you will be soon be shown tricks for performing quickResidual Pipcounts. Anentertaining one (not necessarily fastest) for the above position couldbe:
"Remove aBlack 0:2 and a Red 1:2 checker. Stack Black's 4pt checkers onto the 3pt, and demote his back anchor onepoint to compensate. This sets up ahorizontal symmetry vanishing all 14 checkers which remain on the far side ofthe board. Now, Red's midpoint isworth one of Black's 3 pt checkers, and the other three make 1 (s)quad, with 3 pips left over, for a total of7:3"
Thisillustrates a good example of forcing a symmetry within our grasp. But I offer this just for a taste. You may wish to check it out, blow byblow, after you have read the Residual Pip section.
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CUBEDECISIONS based on Quad Comparison
We have seenhow the Quad Comparison can affect what kind of a checker play we mightselect. Now let's look at the otherreason for comparing Quads: Todecide whether to double, redouble or take a cube.
To this end,we will analyze a position from Jerry Grandell's Important Matches(Ortega/Kleinman), p. 194. (This isa worthwhile book, in spite of the fact that I am in there as one of Jerry'svictims).
Compare yourQuads, and then if you think a Top-Heavy Adjustment is a good idea, you canpractice that too. Write down youranswers, and then check to see if you arrived at the correctcount.
Diagram F
DiagramF -- Comparison:
� Quads: Cancel Red's three Q3's with 9 BlackQ1's -- his extra 6 point checker, and the 8 on the outside. Red's 2:5 ("17pt") checker offsets Red'stwo Q1 checkers. This leaves onlyRed's two midpoint checkers, which count for a 4 Quaddeficit.
� Top-Heavies: Black has 4 Top-Heavies, but Red has 5(one extra). Adjust Black's Quadlead to 5.
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I noticedRed's checkers on the 5 point, and then some high points in the outfield. In addition, I felt a rough Quad countwas insufficient information. A4-Quad lead here made this position a solid take in my mind, but more Quads, Iwasn't so sure; and less quads, then I wasn't sure about the double. The presence of high-pointers and cubesensitivity clearly indicated the necessity of a Top-Heavyadjustment.
Knowledgethat Black's lead is actually around FIVE Quads is enough to give me completeconfidence I should double, and redouble. It is even enough to lean me towards passing. The few immediate possibilities ofhitting Black are ameliorated by the two blots in Red's board, and the holdinggame equity appears insufficient at that big a racingdeficit.
In theactual game, Black did not even proffer an initial double. Perhaps he was operating on a certaingeneral principle which advises, when bearing in against a semi-primed holdinggame, to get the straggler home and lose the market small. I have to wonder, though, if Blackcounted the race.
For example,let's advance Red's two checkers on the 2:5 ("11") point to the 4 point. This makes Red's board more powerful forlater, yes, but, more crucially, for any immediate hits. However, in these key variations, shealso hits less often (198 vs 296 in 1296), so all is largely offset. The main factor of moving Red's checkersforward is a gain in the race, a guarantee to be gammoned on the run less often,and to win the race more often.
Thisalteration slices Black's racing lead from 5 Quads to 3, turning a questionabletake into what is not even an initial double. Such a sizeable swing in cube strengthillustrates how important it can be to refine the straight Quadcount when oneseems to be in an uncertain range. And, of all possible contact positions, a simple holding game, which thisdiagram is rapidly approaching, is the main candidate for cube sensitivity basedupon race.
Thus, it isquite conceivable that one might feel a Residual Pipcount is in order -- atie-breaker to decide a close pass. This time, the RP count below uses squad comparison, which you canprobably already follow. Anyway,you will have a chance to peruse the myriad of Residual Pip cancellation optionsin the upcoming section.
ResidualPips: Black's "Force" near the midpointscancels Red's "Double-Wide" there. Black's Duck, Wide (from 2:5 and bar point), and inner Pair cancel Red'sBig Triangle (home board). Finally,Black's remaining bar pt blot cancels Red's ace pt blot, leaving Red with threecheckers on the 3:3 ("21") point, which count as a Quad + 3 pips. Added to the original 4-Quad comparison,Black's exact lead is 5:3.
In summary,there were 9 pips more which were unaccounted for by the original Quadcomparison, a bad miss. Inrecognizing the 5 and 4 points looked heavy, we opted for a quick Top-Heavyadjustment, which caught 6 of them. Performing a Residual Pipcount instead, though a longer procedure, wouldhave caught that 6 plus 3 more. As"luck" would have it, these 3 extra pips turn out to be enough to nudge thisholding game past the take point.
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ComparingResidual Pips
There aretwo main reasons that you might wish to compare ResidualPips:
(A) You feel that an exact comparison countof a Contact Position might actually swing a checker play or cubedecision.
(B) Your intention had been to spot-check acube decision (no re/double vs re/double, or take vs pass) in a straight race(or light contact position), and you chose Comparison counting over Totalcounting because it is faster. Younow realize that a Residual Pip comparison may confirm the race to be closeenough to Quadcount the leader, and apply the raceformula.
Basically,speedy Residual Pip comparison comes down to the ability to quickly recognizecancellation possibilities. Hereare some options:
� (1) You can cancel Squads on one side withsimilar or different Squads on the other side. As you are ridding these for both colorsat once, it is more orderly to try for the same quadrant, or adjacent quadrants,and handle next whichever checkers remain in that area of the board, ifpossible.
� (2) Checkers on the Acepoint,Barpoint, Midpoint, and/or Roof ("4:1") are called "Aceys". Counting one pip each, Aceys are veryflexible for offsetting, and not just each other. Two of them will cancel a checker anx:2, three of them an x:3, or a 4 and a 1 can cancel a 5, etc. Aceys can easily pair with (or offset)checkers left over from squad transactions, combine with a plus pip-shift, orcounter a minus pip-shift.
� (3) Any checker in VERTICALopposition to an opponent checker can be offset. For example, if both sides have a bloton the 4 point, or both own the 2:5 point, these cancel. (This is the only type of symmetry whichalso works in classical comparison counting).
� (4) Checkers can be offsetHORIZONTALLY, symmetrically around the bar. Black's 3:2 ("20pt") anchor will cancelagainst Red's 1:2 ("8pt"). Or 3:4("22pt") offsets opp's 1:4 ("10pt"). A Black ace point checker offsets a Red midpointchecker.
� (5) Checkers can be offset usingINNER symmetry, within a quadrant. A Black 3:4 ("22pt") anchor balancesRed's 4 point right next door, so once again all four checkers disappear. Applying the same principle, Black'sbar point offsets Red's midpoint -- the opposite end of the samequadrant.
� (6) The final symmetry possibility is"HOP-SYM", found by hopping a checker 6 pips and then applying eitherhorizontal or vertical symmetry. Black's 4:0 ("24") point offsets Red's 3:0 ("18") point. Or Black's 3:4 ("22pt") offsets Red's2:4 ("16pt"). This symmetry is thehardest to notice, but can sometimes prove handy.
� (7) If a desired symmetry doesn'tmatch up exactly, you can force it, then mentally SHIFT checkers elsewhere thesame number of pips in the opposite direction (or just add or subtract a pip ortwo from a running count). Or, youcan (for any reason) swap ANY checker, point, or group, with that of theopposite color -- ANYWHERE on the board!
Note that"Quad-hopping" is a combining technique for same-colored checkers, not acancelling technique for opposite-colored checkers. You can, of course, Quad-hop to positiona Black checker, as long as it is symmetry which causes the actual cancellation(see "Hop-sym" above).
Keep on thelookout for old and new ways to combine and cancel. As you add tricks to your repertoire,you will have fewer and fewer checkers or adjustments to "remember", to thepoint that within a few seconds you will just "see" the Residual Pipdifference.
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Fine-TuningChecker Plays
Decisionswhether to expose a blot or not (Safe vs Bold play) are generally more sensitiveto the exact pipcount than are some of the less volatile types of Aggressive vsPassive plays (such as the 6-4 play in Diagram E). A small race adjustment can swing a playmore easily.
With this inmind, our final example is an early middle game, from Backgammon(Magriel), p.217 [reprinted in Classic Backgammon Revisited (Bagai),#38], on which to test all three Comparison skills you havelearned.
We areBlack, and have a 3-2 to play. Should we hit, or play safe? Is the race a consideration, and if so, at what race threshold would youguess it becomes correct to switch plays?
For eachposition arising in actual play for which you feel a Comparison count is needed,you do a Quadcount, followed if necessary, by only one (or neither, butnot both) of the Top-Heavy or Residual Pip alternatives. However, for purposes of teaching, I amasking you to practice both full comparison alternatives (Quadcount +Top-Heavy, and Quadcount + Residual Pips), and compare your answers with thosefound below.
Diagram G
DiagramG -- Comparison:
� Quads: Black has an extra checker in Q3, whileRed has his extra in Q1 ("6pt"). Red has a 2-Quadlead.
� Top-Heavies: Black has 3 Top-Heavies, Red hastwo. Adjust Red's Quad lead to 3. OR:
� ResidualPips: Swap Red's 5 point with Black's 3:3 nextdoor. Horizontally cancel one ofthe Black's new 3:2 checkers with a Red 1:2 ("8pt") checker. This leaves Black only with his 2:4checker, counting 4pips. Added to the 2 Quads, thismakes a race deficit of 2:4.
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In the SafePlay vs Bold Play chapter of Paul Magriel's book (which is still often referredto as the "Bible"), the above position is given, with the example of a lone 3roll accompanying the diagram. Therecommendation was to hit, but that is a substantial error, as Bagai points out25 years later, because doing so breaks the 1:2 ("8") point.
Hitting withthe 3 reflects the perceptions and priorities of the time. Strong player's of the 1970's battledover key points aggressively, with little concern for the race. There was an unspoken agreement that theonly skillful win was one in which a pure position triumphed over a cracked one-- and especially deserved if one first maneuvered into a backgame orbeautifully complex holding game, well down in the race. It was a charming era, to besure.
I know; Ilearned in that environment. I evenunwittingly helped perpetuate the myths. They/we played these poor positions quite well, and even got away with ithandsomely, until the level of play improved. Eventually, opponents no longer buriedearly when they could slot, or took crushing redoubles. Nor did they go to the other extreme ofdangling so many carrots that it became hilariously profitable to abandon abackgame, or pass bluff re-whips.
Fortunately,we can still pay homage to this biblical archive by altering the roll to3-2. I will assume there aretwo choices: (a) Hitting from 1:4,or (b) Covering 1:4 and playing up to 3:4. (I'll rule out coming down with the 2 and hitting, which leaves fourblots, though it is an "opportunity" at which many '70's players might haveleapt).
Perhapsthere are players who would hit with the 3-2 even if way ahead, and others whowould play safe even from way behind. For those players, counting would be irrelevant. But most players would probably have some race threshold (either exact orrough-range) that would help decide the hit. Consulting the race is likely tohelp.
It turns outthat in this position, because we are down 2+ Quads, it is very right tohit with the 3-2. If wereduce our race deficit one Quad by advancing our anchor out to the bar point,then hitting is still correct, but not to as great a degree. As we inch our anchor towards themidpoint, the margin narrows by about 1% per pip pair (there is a safe-warp at the 16 point, but that hasto do with the value of using the 2 to connect the back checker), and by thetime we get to the 14 point, it is nearly a tossup. With the aid of rollouts, we havediscovered the threshold: We shouldhit if down 2 pips or more.
If wefurther advance one of these checkers to the mid and the other to the "6" point(to give us a 1 Quad lead), we arrive at Magriel's counterdiagram, and, as youwill guess by now, playing safe is (very) correct even though we can hit withoutbreaking the 1:2 ("8") point. It isa pity that Paul did not either choose our 3-2 roll, OR stick with the lone 3but add a builder to the 1:2 point (from the mid); either way, that diagram pairwould have supported his well-conceived theme perfectly. It is also a pity that Paul is such anhonest fellow; otherwise he could claim his diagrams weremisprints.
Contrary topopular belief, the issue of having more versus fewer checkers "back" playssecond fiddle to the race as a criterion for choosing a Safe or Bold play. Having established the 2:2 ("14") pointas a Safe/Bold threshold: If, fromthere, we move two checkers from Red's mid to her "6" point (to refund the 14pips we stole, in the way which least affects immediate tactics), hittingis correct by 7%, just as in the original diagram. In spite of the fact that Red isnow the one with the extra checker "back", we make the bold play because, asbefore, we are down 2+ Quads.
That wemight not know that the Safe/Bold threshold is -2 pips (or even in theneighborhood of zero Quads), is no excuse for throwing up our hands and notcounting the race; a better informed decision will be right more often. A more rational excuse is that we areslow at counting, and don't feel it is fair to keep our opponent (and perhaps byripple effect, half the people in the tournament hall) waiting. I applaud that "excuse"; hearing it islike a breath of fresh air. Butdon't let me get started on that. It suffices to say that if we get faster at counting, we no longer needto be a slave to that excuse, nor to chugging away at tediousarithmetic.
Quadcountsare off about 4 pips on average (though can stray a fair bit further). Will this inaccuracy make a differencein the play that we select? Sometimes.
If we feelthat our decision is race-sensitive and the Quadcount is on the bubble, then wecan add in the Top-Heavy Adjustment, which is quick and cuts the inaccuracy inhalf. Or, if a Safe vs Bold Playposition is close to a reference position we've seen, and/or we are capable ofmaking sound adjustments, then the pinpoint accuracy of a Residual Pipcomparison could well determine our choice of play.
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RacingOnward
Don'tworry! You are not expected toremember all of the counting nuances in this article. If you have even followed, let aloneabsorbed, ten percent, you are doing very well. The numerous options are not meant tooverwhelm you, but to be your friend, so add what you like, a little at atime. Meanwhile, Naccel canfunction quite adequately on the mere fraction you havedigested.
One of themajor ways we improve our level of playing skill is to build up a portfolio of"reference positions" (organized in a notebook or loosely etched on our brainsover time) which indicate exact (or rough-range) thresholds at which we shouldalter certain checker plays or cube actions. While the race is only one contributingfactor in making these decisions, it is a more dominating one than players inthe past have realized.
Backgammonis, in essence, a race. The respectmodern players have gained for the race is reflected in the types of positionsfor which they strive, and in their inclination to count a greater variety ofpositions. For this purpose, itmakes sense to choose a method of counting that is not only multi-faceted,flexible, fast and reliable, but also fun. It is my hope that, by adopting Naccel, you will be encouraged to countthe race far more often, while investing less overall time doingso.
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If you wouldlike to continue practicing your counts, verification of accuracy is much easierif pipcounts are pre-posted next to the diagrams (to convert, multiply yourQuads by 6 and add the residual pips). Suggested sources are: TheWoolsey and Bagai (only 3 count errors each -- can you find them?), andOrtega/Kleinman books, from which this article borrowed diagrams, and MarioKuhl's magazine Backgammon Today. Finally, if you save online matches, every move gives you a new position,which you can count and then verify with the touch of abutton.
Our deepappreciation goes to Ric Gerace for providing such legible color diagrams, andincorporating so precisely the new quadrant numbers, point numbers, and quadrantdivisions. (You can find some coolstuff on Ric's website, which is http://www.ricgerace.com/).
This is thefull-length August 2001 version of the Naccel article. A slightly updated, though edited,version will appear in September in BackgammonToday.
Ulf Wostneris planning to construct a website, for the purpose of teaching Naccel. If successful, this article will bereprinted there, in expanded form, along with reader input and various teachingmechanisms, gradually improved over time. The address is www.cyberprof.com/nack, though we do not know how soon itwill be functioning.
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