Naccel: a[N accel]erated Pipcount


by Nack Ballard


Count -- On Me?


"Twelve plus fifteen plus six in my board, that's thirty-three, plus sixteen for the two on my eight point, that's forty-nine, plus nine equals fifty-eight, and fourteen equals seventy-two, plus three on the... what IS that, the 'twenty-two' point, is sixty-six, added to.... Oh no, what was it again?..."  I mumbled aloud.


"I wasn't really paying attention... sixty-six?" my doubles partner suggested dubiously.  "You had just counted the checker on their eleven point", he added helpfully.


"Well, yeah, thanks, I know that... but it's the ANCHOR that is sixty-six.  I forgot the running count I was supposed to add it TO..."  Here I go again, I thought... now I'll have to start all over again for the tie-breaking count.  "What were my first two tries?"


"Hmm....they differed from each other by eleven... I remember that..." his voice faded as quickly as his grin.


I could already feel my equity shrinking.  "Well, let's start again with what we know.  What did we get for the count on their side, 130-what?"


"I know it was my JOB to remember the counts", he savored the word like a sour plum.  "But when you asked me about your last pipcount, it all went out of my head."


"I wasn't asking you, I was asking myself".  Hmm, that didn't come out right.


"Well, my ability to distinguish rhetoric isn't what it used to be, nor is my memory.  I can't even recall why I paid both halves of our entry fee."


I was no longer sure myself of the reason.  I wanted to quip back on his misuse of the word "rhetoric", but someone had to pull the team back together.


"Okay, LOOK..."  Pretending I was still in charge was the only card I had left to play.  "You count Red and I'll count Black; we'll do it really carefully this time, and whatever we get, we'll just go with THAT."


... Sound familiar?  Well, maybe your presence of mind is not quite as absent as mine, but might not such a calamity occur if you find yourself deprived of enough sleep, or otherwise lacking focus?


Counting pips costs time, and drains our reserve of energy.  It is important to be able to stay intent on the positional considerations, which, if assessed correctly, will more likely guide us to correct plays and cube decisions.




Following a Dream


With the good of the human race in mind, then -- well okay, I admit it was because I wearied of my own incompetence -- I actually dreamt (literally) a new method of pipcount.  In my dream, I was playing on Gamesgrid, clicking on the pipcount button (that wonderful online crutch), when out popped a cyberboard with the four quadrants taking turns bulging out at me.


The next morning, while driving, I mused how easy counting the race would be if all I had to do was count FOUR BIG points instead of 24 little ones.  No more two-digit numbers to add or multiply.


Thinking further, it seemed as if this simple scheme would indeed give a good approximation.  Beginning players could benefit by being taught to initially adopt a Quadrant Count, to get a feel for how much they were ahead or behind.


Then it hit me.  An expert player could use this absurdly simple counting method, too.  The only other step would be to count the remaining pips in each quadrant!


When I arrived home, I set up some backgammon positions.  To the checkers in my home board I assigned a quandrant value of 1, to those in my outer tables 2 or 3, and to my back checkers 4.  Counting the total quadrants was as easy as counting a bearoff in which my checkers were all on the 4, 3, 2 and 1 points.


To do full pipcounts, I altered the designations of the quadrants (from 1, 2, 3, 4) to 0, 1, 2, and 3, in order to allow residual pips to be ADDED instead of subtracted.  (A checker on the "22 point" needs to travel THREE (not four) quadrants in order to bear IN, and then 4 more pips to bear OFF).  Hmm... even easier!


I started leafing through any backgammon literature I could find which had pipcounts accompanying the diagrams.  At first, I made a lot of silly mistakes, overlooking leftover pips.  But, as I kept practicing, and gradually streamlined the method (in particular, after introducing "squads"), I was astounded to find myself rattling off correct pipcounts in a matter of seconds.


A wave of euphoria, bottled up by 25 years of slavish pipcount, washed over me.  I shared the details of my discovery with a friend, Ulf Wostner, who mirrored my enthusiasm by offering to set up a website (see the end of this article) to teach this new method of counting pips.


It was Ulf, in fact, who convinced me to dub this system "NACCEL".  I have to admit, I like it.  As I have no children, perhaps I imagine this appellation as some alternate way of spreading my genes.  I think it is only fair to clarify that I attribute neither my psychological imbalance, nor my need to compensate, to any lack of attention from my parents or teachers.  I realize if I were a more naturally modest, decent fellow, I would have insisted on my original name of "Accelerated Pipcount" or even "Speed Counting".  But it is too late to pretend.  Anyway, take your pick.




Other Pipcount Methods


The most common way to count total pips is by the "Straight Count".  The number of checkers on each point are counted, weighted by their point numbers from 1 to 24.  If there are two checkers on the "23" point, they count as 46.  Three checkers on the "14" point count as 42, and so on.  These weighted subtotals are added together to determine the total number of pips necessary to bear them all off.  To function as a human calculator can be tedious, though the process does get easier as one gains familiarity with multiples of the point numbers which arise most often.


Jack Kissane's "Cluster Counting" improves on straight counting by isolating commonly found checker clusters having pipcount multiples of 10 (or 5).  Mental shifting can produce these clusters (other checkers to be moved the opposite way to compensate), or pips left over are added or subtracted.  Basically, Cluster Counting utilizes several helpful reference clusters with straight counting as a fallback.


There are two other interesting pipcount methods which I heard about only after near-completion of this article: 


One is Mark Denihan's "Quadrant Crossover" technique (outlined in an article by Mark Driver).  Its basic idea is startlingly similar to one of my prototypes for Naccel:  Count the quadrants, multiply by 6; to count the remaining pips visualize the 1, 2, 3, 4, 5 and 6th points stacked on top of each other, and add the six numbers together.  Obviously, since I did not stick to that blueprint myself, I believe Naccel to be a substantial improvement.


The other is Douglas Zare's half-crossover method, which weights 8 half-quadrants ("triples"), and then adds 75 to get an excellent approximate count.  This is a clever idea, as one would expect from Zare, and easier than straight counting.  However, to get an exact count, Zare says it all himself, when he humorously preludes his article with an excerpt from Lewis Carroll's classic:


"And you do Addition?" the White Queen asked.

"What's one and one and one and one and one and one and one and one and one and one?"


"I don't know," said Alice.  "I lost count".




How does Naccel work?


The basic precept for Naccel is that each checker is required to travel through a specific number of QUADRANTS to bear IN, and through a specific number of POINTS to bear them OFF.  One need not identify, let alone multiply, the "19" or "21" point or such -- there is nothing higher than the 6 point!


The opening position for Black's checkers is illustrated in Diagram A below.  Black's Quadrants are marked "0", "1", "2", and "3" and "4" for a checker on the roof.  In the text, we will refer to them as "Q0" thru "Q4".  Just as in the counting of points, each unit is a "Pip"; in the counting of Quadrants, each unit is a "Quad".


Notice that the Quadrant divisions are shifted one point.  This will be easy to get used to, I promise, and the reason for it is simple:  It is more efficient to count 1:0 ("6-point") checkers as a Quad in and of themselves (or 2, 3 or 4 quads, in the case of the "12", "18" or "24" points), rather than as 0:6 (or 1:6, 2:6, or 3:6), which would mean being saddled with six residual pips for each one.


Checkers are pip-defined by the point numbers (0 thru 5) on which they stand -- the number of pips required to bear INTO the next quadrant.  Look at the number AFTER the colon of a notated point; for example, "1:2" has two residual pips.


It is difficult to emphasize strongly enough that becoming accustomed to calling the points by their new names will help you enormously in using Naccel.  For purposes of this article, we will refer to points by their new (mod-6) notation, though we will usually list the traditional point numbers in parentheses as a reference.


For example, what is classically known as the "10 point", is Quadrant 1, Point 4, so we designate it "1:4".  The "23 point" is Quadrant 3, Point 5, or "3:5".  [Note that multiplying the number before the colon by 6 and adding the number after the colon translates back to the traditional number of any point].


Checkers in Q0 will mostly be referred to by their classical name; thus 0:5 can still be called the 5 point.  The "13" point is now "2:1", and the "7" point "1:1", though we will mostly refer to these by their descriptive names, the "midpoint" and "bar point".



          Diagram A

Okay -- How do we count, using these new-fangled point numbers?




How to do a TOTAL Pipcount


                     Step 1:  Count the number of "Quads".

                     Step 2:  Add to that the "Squads".

                     Step 3:  Count any leftover pips.




We will now clarify these steps, as we count Black's starting position (diagram):


(1)   Count the checkers in each quadrant, as if they are on four "big points", in order to determine the number of Quads necessary to BEAR IN all our checkers:


            The 2 back checkers ("24pt") in "Q4" count 4.  (2 x 4) = 8.

            The 5 checkers (on midpoint) in "Q2" count 2.  (5 x 2) = 10.

            The 8 checkers (6 and "8" pts) in "Q1" count 1.  (8 x 1) = 8.

                                                                        [8 + 10 + 8] = 26.


(2)   The three checkers on the 1:2 ("8pt)" are a "Squad" (a 6-pip unit).  26 + 1, that makes 27 altogether.


(3)   Count the leftover pips:  The five midpoint checkers each count 1 pip.  5 x 1 = 5.




I have outlined the count in great detail above, but if you had been able to listen to me count under my breath, what you actually would have heard, in a span of three or four seconds, would have been "8..18..26..27, and 5". 


An Attentive Reader:  But where is the rest of it??


Nack:  "There is no more.  My count is 27:5.  What is yours?


Reader:  "Two on the 24 is 48, plus 13 times...65... a hundred and uh.. 13, plus 24.. 137.. and 30.. 167."


Nack:  "Perhaps familiarity of the opening points helped, and there were only four of them, but still... 14 seconds... you are pretty fast for a straight counter.  And you got the right answer.  But did I notice you splatter a few beads of sweat as you jittered out the arithmetic?"


Reader: (hiding smirk well):  "Well, at least someone has the total now... What good does this 27:5 do you? 


Nack:  "Technically, it means 27 Quads + 5 pips.  I am confident that if I multiply the 27 by 6, and add the 5, I will arrive at the same total you got.  But I never have the need, as you will see".




Counting QUADS


This stage is quite simple.  Imagine a SPEEDBOARD.  That is the equivalent of what you are counting.  Your quadrants are just big points, counted as 4, 3, 2 and 1.


The roof is your "4 point".  The opponent's home board, your "3 point".  The opponent's outer table, your "2 point".  Your outer table, your "1 point".  Your inner board (below the 6-point) doesn't count at all.  So, overall, you have the equivalent of a speedboard with fewer, sometimes far fewer, than 15 checkers!


I admit, at least at the outset, that counting will not be as simple as this analogy suggests.  Because the quadrants are shifted one pip, the bar (and board edges) visually hinder the "6", "24" (and the less seen "18" and "12"), point checkers, from being assigned to the proper "points" of your speedboard. 


I can assure you, however, that with a basic understanding of the system, and practice, any such confusion will rapidly disappear.  You will find ways to remember; for example, it is easy to toss the "6" point in with the rest of the Q1 checkers when it is its usual conspicuous, towering self.


Perhaps the other imperfection in the analogy is that checkers spread out over an entire board are not side by side, as they are in a speedboard, so your eyes have to travel a bit further to count.  What you pay for in distance, though, you are sometimes more than refunded in compactness.  For example, it is faster for me to count an outer 4-prime as 8 checkers, than it is to correctly count 8 checkers on the 1-point.  There is not a big difference either way -- the ability to count the raw number of checkers in a quadrant at a glance is, as you might expect, just a matter of practice.


In learning to count your Quads, it makes sense to start on one end and sweep systematically; in this way you can be sure to correctly account for every checker.  I recommend starting with Q4.  Just as you would with a speedboard, count your big "points" (quadrants) first.


As you gain confidence, you can improve your speed by selectively combining checkers in different quadrants in the same way you might combine different points of a speedboard.  For example, you know that two checkers each on the 3 and 2 points create a "block" of 10 pips.  In the same way, two checkers each in Q3 and Q2 create a 3322 "block" of 10 quads.  (Hint:  The midpoint checkers are the most common Q2's, by far).


One Q-pattern that frequently arises is a checker on the 4-0 ("24") point (or roof), plus an anchor (or two checkers split in the opponent's home board).  This 433 pattern also counts as 10 quads.  Or, the anchor can instead be three on the midpoint, say, to create another useful 10-quad pattern of 4222.


Five checkers in Q2 (e.g., on the mid), or 22222, will become as obvious a 10-quad pattern as five checkers on the 2-point is a 10-pip pattern.  Eventually, you can introduce other patterns, such as 222211, or 2221111 (usually combinations of midpoint and "8" point checkers), which occur to you, and the amortized gain will save more time than the initial cost in finding or remembering them at the board.


If 10-quad patterns don't fall into your lap, don't fret.  Positions generally range from 12 to 25 total quads, so your options of extracting a convenient block may be limited.  Just retain that image of a speedboard in your mind.  You might be drudging along, counting quadrant by quadrant, until it occurs to you, say, that four on the "2-point" plus four on the "1-point" are like having four on the 1+2 "point".  Similarly, four Q2's plus four Q1's count as 12.  Whatever the pattern is, once you discover it and decide you like it, you can add it to your repertoire.


Soon, we will describe the other half of the equation.  Then, we will put the two halves together, and practice full board counts.




Origin of a New Term


With Quads quickly counted, what is left?  The answer is:  All the pips which checkers require to bear INTO the next quadrant. 


My first system was to scrape these residual pips together into patterns of 10, but eventually it dawned on me that I could allow the geometry of the backgammon board to work in Naccel's favor for this phase of the counting as well.  I have found 6-pip checker groups to be prettier, more manageable, and fit much more naturally onto the board than the 10-pip groups I have subsequently discarded.  Most importantly, the use of 6-pip groups has meant that EVERYTHING can be converted to the equivalent of Quads, with a mere handful (literally, 0-5) of leftover pips.


I wanted to think of a catchy name for these 6-pip checker patterns.  "Clusters", which is a more aesthetic name than "Clumps" or "Combos", was already taken, and "Groups" or "Sets" seemed mundane.  "Virtual Quadrants" was descriptive and catchy, but a bit long, and "Virtuals" sounded funny.


The next try was to find names that suggested the number six.  Looking for something jazzier than "Hexads", I got sidetracked to mental depths I barely dare to repeat.  The trouble began when "Sex", the Swedish word for six, was suggested to me, and "Sexes" became the prime candidate.  Individual names for "Sex" shapes filled my brain:  Checkers in a block shape became a "box", checkers in a stack a "rod", checkers along the edge a "lay", three on a point a "three-way", and so on.  I even had an animated argument with myself about whether a certain shape looked more like a "spoon" or a "sucker".


It is a matter of definition whether I ever recovered my sanity, but it was when the word "Squad" suddenly struck me, that I was jolted out of my Scandinavian fantasy.  A term that suggests tactical deployment upon a battlefield, "Squad" gives life to the checkers, "men"; yet it literally means a group of "people" -- hey, that might even please the feminists (if such a thing is possible).  Also, quadrant rhymes with squadron (close enough), and Quad with Squad.  Clearly these blood brothers are meant to be a tag-team; it seems only natural for "Squads" to take over where "Quads" leaves off.




The Ten Basic SQUAD Patterns


The use of "Squads" is nothing more than a system by which to quickly and conveniently count the Residual Pips.  This would include each of the checkers on the 6, 1:6, 2:6 and 3:6 points, except that we had the foresight to shift our quadrant boundaries (one point), thus including them in the Quadcount.  We need only Squadrify the remaining checkers, pip-defined by the point 1, 2, 3, 4 or 5 on which they stand.


A "Squad" is, basically, any six pips.  The Ten basic Squad patterns are incorporated into Diagrams B1 and B2 below (with a few irrelevant checkers on the 1:0 ("6") points so that you can get used to ignoring them):



          Diagram B-1


          Diagram B-2


Exactly as in the traditional notation system, everything is reversed from Red's point of view.  To get the Red notation for a point (until it becomes second nature), you read the point on the exact FAR SIDE of the board (just as you must do with all classical diagrams).


Translated into shorthand, the Ten Basic Squads are:  33, 42, 51, 222, 321, 411, 2211, 3111, 21111, and 111111.  A name and description of each follows:



                     (a)  33.  The "PAIR".  Starting with the most obvious pattern, two checkers on the third point of any quadrant add up to 6 pips.  This appears in the top diagram as Black's 3:3 ("21") point, and in the bottom diagram as Red's 3 point.


                     (b)  42.  The "SPLIT" can be thought of as (4+2) = 6, or as two checkers which can be shifted, one a pip forward and the other a pip backward, to form the 3 point.  In the top diagram, Black has a 42 in his home board.  In the bottom diagram, Red's back checkers form a 42.


                     (c)  51.  The "WIDE" (as in wide split).  See top diagram, Red's bearoff.  Again, a mental shift will transform these two blots into her 3 point.


                     (d)  222.  The "DUCK", appears as 3 checkers on Red's 1:2 ("8") point in both diagrams.  Although Q1 is the most common place to find it, 222 can appear in any quadrant.  A "duck" is a short stack of stones marking a cross-country trail.  In backgammon, "ducks" refer to deuces.


                     (e)  321.  The "LAYER":  See Bottom diagram, Black's Q1.  It is easy to see this is the bottom layer of a 321 prime (the "Double Layer").  Whenever we add a layer, we add a unit of Squad.  Other Squad patterns can be easily visualized in combination with (on top of) the Layer.


                     (f)  2211.  The "BLOCK".  See Top diagram, Q1, or bottom diagram, Q0, both Black.  This is a really handy formation.  If doubled in height, it becomes the "Building" (or if tripled, the "Skyscraper").


                     (g)  411.  The "WEDGE".  Bottom diagram, Black's Q2.  As with all basic patterns, these can be combined from separate quadrants; e.g. 3:4 ("22pt") + two on the mid.


                     (h)  3111.  The "TRIANGLE":  Top diagram, Black's Q2.  A good resource for a 3-point checker which can't form a pair, or find a layer.


                     (i)  21111.  The "SOCK":  Top diagram, Red's Q2.  Useful at the end, to round up 1-pointers (midpoint, bar pt, ace pt, roof).  See also the "Stack".


                     (j)  111111.  The "STACK":  Bottom diagram, Red's Q2.  Six checkers on, or symmetrical around any point, convert to an exact number of Squads (in this case, 1).  For example, six checkers on a 5 point count 5 Squads.




Multiple Squad Patterns


The most useful Multiple Squads are "isolated" -- those which cannot be constructed by piecing together Singles.  That these Multiples are the Singles' mirror images around the 3 point helps to reinforce the patterns.  (The Pair, Split and Wide above, and the Kicks below, are their own mirror images).  Multiples arise less often than Singles, but are worth their weight in gold because they combine the otherwise clumsy checkers on the 5 and 4 points.


The isolated Doubles are 552, 543, 444, 5322 and 4431.  They are illustrated, along with the two Triples of 5553 and 5544, and the one Quadruple of 55554, below in Diagram B-3. 


The Quintuple-Squad 555555 ("Big Stack") is the mirror image of one of our basic Squad patterns (the 111111 "Stack"), but is so rare, that I have not depicted it.



Diagram B-3

                     (a)  552.  The "FIN", or "Big Wedge" is in Black Q1.  A "fin" is jargon for a five-dollar bill, and this pattern also resembles a shark's fin.  A theme in all multiple-5 formations, The "2" gives one pip to each of the 5's and tips you off that the formation counts 2 Squads.


                     (b)  543.  The "BIG LAYER" (2 Squads), is the mirror image of the 321 Layer (and the same ideas for building up apply).  If you look at Black's home board, you will see a Big Layer partially lurking under other checkers (as layers often do).  Remove it, then take away a "Fin", and you will see that the home board is left only with a "Wide" (51), which comes to a total of Five Squads.


                     (c)  444.  The "FORCE" ("May the fours be with you is a popular backgammon pun"), or "Big Duck" (2 Squads) is in Red Q0.  It is easy to combine checkers from different quadrants, e.g. a 1:4 ("10-point") blot hops forward exactly one quadrant, or over from the 3:4, to stack onto the four point.


                     (d)  4431 and 5322.  The "KICK" (looks like a foot kicking a soccer ball) -- see Red's checkers in both outer boards.  This is the only pattern whose mirror counts the same (2 Squads), hence only one name.  An easy shift (one checker back, the other forth, one pip) transforms either Kick into the 4422 "Double Split".


                     (e)  5544.  The "BIG BLOCK" (3 Squads) is in Black's Q3.  The mirror image of the "Block" (2211), 5544 is great for ridding the least combinable checkers.  If twice as high, it is called the "Big Building".


                     (f)  5553.  The "BIG TRIANGLE" (3 Squads; the "3" part tells you so).  In Black's home board, this can be removed to leave 5421, which I call the "Split Layer" (a combination of the 42 Split and the 51 Wide); it adds 2 more Squads, for a total of 5 Squads in the home board.


                     (g)  55554.  The "BIG SOCK" (4 Squads, as the "4" indicates).  Of all the ways we have counted this particular Black home board, using the Big Sock is easiest.  Left over is simply a Layer (which again brings the count to 5 Squads).


"One even, two odd" is a useful rule to apply to all formations, including the Squad Combinations below:  When there is one blot (or a lone checker on top of a primish squad), it is always on an even-numbered point.  If there are two blots, they will both be on odd points.




Squad Combinations


Most multiple Squad patterns are combinations of two or more Single ("basic") patterns.  Consider these advanced; knowledge of them is a lower priority than of those illustrated in the "B" diagrams.


It is great practice to visualize (or set up) the following combinations, and determine which single (or multiple) patterns they combine.  (Note how the mirror images help to reinforce the patterns).  If you are not in the mood to do so now, feel free to skip this section and come back to it later, or use it as a reference.


                     2 Squads:  4332 = "Hat".  54111 = "Wide Wedge."  441111 = "Double Wedge".  44211 = "Drop-kick".  43311 = "Tandem".  4422 = "Double Split".  5511 = "Double Wide".  5421 = "Split-Layer".  332211 = "Double Layer'.  33222 = "Chair".  3222111 = "Boot".  322221 = "Top Hat".  333111 = "Odds".  22221111 = "Building".


                     3 Squads:  333222111 = "Triple Layer".  433332 = "Big Top Hat".  44433 = "Big Chair".  4433211 = "Truck".  443322 = "Tri-pair".  444222 = "Triple Split".  555111 = "Triple Wide".  55521 = "Wide Fin".  553311 = "Short Odds".  55332 = "Big Tandem".  55422 = "Big Drop-kick".  554211 = "Feet-In".  544221 = "Feet-Out".  543321 = "Sombrero".


                     4 Squads:  555333 = "Big Odds".  555531 = "Wide Triangle".  544443 = "Giant Top Hat".  5543322 = "Big Truck".  55433211 = "W".  444332211 = "Fourplex".  554433 = "Double Big Layer".  55442211 = "Split Blocks" or "Double Split-Layer".  [8-4] Prime = "Special 5-Prime".


                     5 Squads:  5554443 = "Big Boot".  554433222 = "Big Fourplex".  The [1-5] Prime or any "Six-Prime" also counts 5 Squads.




Squadcount Trouble-Shooting Guide


This is the final preparation for counting full boards (and also a reference for later).  A comprehensive look at the pitfalls which are possible during a Squadcount will prevent you from being led astray:


                     (1) Far Side confusion:  This is the most likely cause of a miscount.  There can be a tendency, at first, to confuse the middle points in Q2 and, especially, in Q3.  This is because Red's 2 and 3 points are Black's 5 and 4 points, and vice versa. 


Remedy:  Until instant recognition sets in, just keep reminding yourself which direction the checkers are traveling for the side you are counting.


                     (2) Shrinkage:  Uncounted checkers disappear as if they were counted.

                     (3) Ghosting:  Checkers you have taken off the board reappear, causing you to combine/count them again. 


Remedy for both:  Sweep systematically, as if you are vacuuming a rug, so that you are less likely to forget or redo a corner.  Avoid darting around, leaving isolated checkers or groups.  If you collect only part of a point, "vacuum" the rest of it as soon as possible.


                     (4) Pattern Confusion:  2111 is too short to be a sock.  432 is a bogus layer.  441 is not a real wedge.  5422 is a footless kick.  3322 or 4433 is a phony block. 


Remedy:  Verify that patterns total 6, or a multiple of 6.  Review Diagrams B1 thru B3, and keep practicing your counts.  Getting these patterns right is mainly a matter of repetition; soon, the misfits will just plain look wrong.


                     (5) OverSquads:  With proper combining, these rarely arise, but, occasionally, your final remainder will be more than 6 pips, with no squads available.  These rogue combinations are 322, 431, 441, 443, 522, 532, 553, 544, 5554, 55555, and certain subsets.


Remedy:  Review the Multiple Squads section, and focus on combining 5's and 4's first.  Ration low point-count checkers (e.g., midpoint and 1:2).  If you do oversquad, just shift the high checker(s) up to the 6 point, to create a 1-checker squad, or shift to make the 3 point.  For example, 553 or 544 can shift to 661 = 2:1, or 443 becomes 533 = 1:5.  This is a one-time deal, and occurs last; thus, it is easy to add to your (s)quad total.


                     (6)  Crossover Shift:  What appears to be an innocent shift between the "6 point" and a lower point, actually crosses over a quadrant boundary.  (By contrast, note that shifting 1:0 to 1:1 ("6" to "7"), or vice versa, is fine, because that does not cross over).  Similarly, with the "24", "18" or "12" points, though the temptation to shift there is relatively infrequent.


Remedy:  Make your shift BEFORE counting the original Quads (and then so as not to forget, start the Quad count from that end).  If you do shift after, subtract 1 Quad if shifting 6-5 (or add a Quad if 5-6).  Or, sometimes you can plug the desired prime holes by "Quad-hopping" (shifting checkers 6 pips), which can be done freely, without burdensome side effects.  The text accompanying Diagram D contains illustrations of this theme.




QUADS and SQUADS -- Putting it All Together


Yippee!  It is time to apply the techniques we have learned to count full boards. 


Take another look at Diagrams B1, B2 and B3.  Count the Quads, and then add the Squads, as you go, to the Quads (sub)total you got.  Do not convert to pips.  Write down your Quad totals for Black, for Red, and the Quad difference.  Then compare them with the answers, below.


[Please note that the term "Quads" refers both to the quadrant units first counted, and also to the combined total of these quadrant units and the Squads added to them.  In context, it is usually easy to see which one is being referred to, but if there is possible confusion, then the terms "Original", or conversely "Combined" (or "Total") can preface].


If you feel slow, or lose track of the Naccel procedure, or get the wrong answers, review "How to do a Total Pipcount", "Counting Quads", or the Squad sections, as needed, and try again.




Diagram B-1 -- Black:


Quads:  The anchor (which is two checkers in Q3) counts as 6 quads.  The clump of 4 checkers in Q1 is convenient to add to it, because that makes 10.  The three on the 1:0 ("6-pt") are also Q1 checkers (remember), so that's 3 more quads, making 13 so far.  The four checkers in Q2 count 8, for a total of 21 Quads.


Squads:  Starting from his back checkers and sweeping around:  Black has a Pair, a Triangle, a Block, and a Split, for a total of 4 Squads.  Adding that to the 21 original Quads, makes 25 Quads altogether.


Diagram B-1 -- Red:


Quads:  The five in Q2 are nice, that makes 10 Quads.  There are 8 more in Q1, for a total of 18 Quads.


Squads:  Starting from the midpoint area:  Red has a Sock, a Duck, and a Wide, for a total of 3 Squads.  That makes 21 Quads altogether.


Summary:  Red leads 21 Quads to 25, a difference of 4 Quads.



Diagram B-2 -- Black:


Quads:  Black has three in Q2, which count as 6 quads, plus 8 more in Q1, makes a total of 14 Quads.


Squads:  Black has a Wedge, a Layer, and a Block; that's 3 Squads.  That makes 17 Quads altogether.


Diagram B-2 -- Red:


Quads:  Red's back checkers count 6, the midpoint is 12, plus 5 in Q1, makes 23 Quads.


Squads:  Red has a Split, a Stack, a Duck, and a Pair; that's 4 Squads.  That makes 27 Quads altogether.


Summary:   Black leads 17 to 27, a difference of 10 Quads.



Diagram B-3 -- Black:


Quads:  Black's back checkers (four in Q3) count 12, plus 3 in Q1, makes 15 Quads.


Squads:  Black has a Big Block (counts 3), a Fin (2), a Big Sock (4), and a Layer (1), for a total of 10 Squads.  Added to the 15 Quads, that's 25 altogether.


Diagram B-3 -- Red:


Quads:  Four Q2 checkers make 8, plus 8 in Q1, makes 16 Quads.


Squads:  Red has two Kicks (or two Double Splits by virtue of shifting) -- each counting 2, so that's 4 so far.  The Force in the inner board counts as another 2, for a total of 6 Squads.  Added to the 16, that's 22 Quads altogether.


Summary:  Red leads 22 to 25, a difference of 3 Quads.




Counting a Real Game Position


Squads are usually not ALL as conveniently arranged as in the B diagrams.  Let us try counting a position which arose in an actual game -- a tricky middle-game example from New Ideas in Backgammon (Woolsey/Heinrich) #21 (p. 60), illustrated below.


Admittedly, we would not count this position in live play.  No matter what we discover the race to be, it is too risky to break the anchor and hit with the 6-3 rolled in the actual game.  Nor can either side consider a double based on the race until there is a shot, serious deterioration, or some sort of contact is broken.  We are counting this position purely for practice.


Starting with this diagram, I recommend you pull out the original books (if you have them), and open to the page from which I've borrowed the diagrammed positions for this article (or print or photocopy from here).  This will avoid having to scroll your screen or turn pages back and forth in an attempt to follow explanations.


One final recommendation, before beginning:  To best benefit from this article, make the effort to understand each adjustment in each step of the full counts offered under each diagram, before moving on.  At first this may seem tedious, but when you catch on, your reward is that your mind will probably feel a bit like a rocketship at takeoff.


Again, count on your own (Quads + Squads, and leftover Pips, please).  Write down your steps, and compare your answers to those immediately below the diagram:



          Diagram C

Diagram C -- Black:


                     Quads:  Two Q3's is 6, plus 12 (in Q1) = 18.


                     Squads:  The 3:3 point is a "Pair", for 19.  The 5544 is a "Big Block" -- that's 22.  The "Duck" makes 23, with just 1 leftover pip (ace point).


Diagram C -- Red:


                     Quads:  Two in Q3 and two in Q2 (the 3322 combo) make 10, plus 7 in Q1 makes 17.


                     Squads:  The 3 point "Pair" -- that's 18.  The 5 point plus a 1:2 checker create a "Fin" (counts 2) -- that's 20 -- which leaves a "Block" outside, that's 21.  The 2:4 goes with one of the 3:2 for a "Split", makes 22, with 3 leftover pips (3:2 + mid).



TOTALS:  So, Black has 23:1, and Red has 22:3.  Red leads by 4 pips.  [If you don't yet see this difference easily, count on your fingers up from 22:3, thusly:  "22:4, 22:5, 23:0, 23:1"].




How did you do?  Your count was wrong?  Okay:  Try to figure out how it happened.  And don't worry.  There's a significant chance that you will make a mistake or two the first few times.


I gave a longer solution in the text above, but actually I lucked out and counted Black's entire position in under 3 seconds.  "6..18... wow -- 23, and 1."  Can you guess what I had done?  I had noticed that Quad-hopping the 3:3 point around and inserting it into the 1:3 slot formed the "Big Fourplex" pattern (worth 5 Squads).


Before long, squads will jump out at you right and left, even when combined from different quadrants (believe it or not), and you will be counting them as confidently as chairs around a table.  There will be many choices, but finding squads which use up the 5 and 4 point checkers first will retain more flexibility for coralling the rest of the checkers.  You can let that principle guide your sweep, as it did Red in the above position.


Red went straight for the 5 point (knocking off the Pair on her way, so as not to leave it isolated), and had a choice of 2-point checkers to combine.  She chose the one outside because it left a Block there, but she could just as easily have grabbed one of the 3:2 checkers, combined the remaining one with the 2:4 (a Split), and shot down the Duck, with 3 one-pointers left over (bar point and midpoint). 


An experienced counter would likely see Red's 3:2 point as 4 pips to be combined onto the 2:4 blot, and quad-hop them to the 4-point for a "Six-prime" (5 Squads), with 3 pips left over.




Counting the RACE


We utilized the last diagram mainly to practice a full count.  In actual play, the only good reason for needing to know the TOTAL count for both sides (as opposed to just a Comparison count) is in order to decide whether to double, to redouble, or to take or pass a cube in a straight race (or light contact position).


Consider the position below.  Should Black double?  Redouble?  Should Red take?


Let's start by determining the total count (Quad + Pip format) for both sides.  Do NOT convert the Totals to Pips, only the difference to Pips.  Don't worry:  In the next section, I will show you how to make accurate cube decisions with the same Naccel-style numbers we have been producing, and with greater ease than you have ever experienced with straight pipcounts!



          Diagram D

Diagram D -- Black:


                     Quads:  Two Q2's (midpoint) is 4, plus 10 (in Q1) = 14.


                     Squads:  The 4 point + 1:2 ("8") point (a great "Double-Split" to know), makes 16, plus a "Layer" makes 17.  The 5 and 1:5 team up with the midpoint for two Wides, making 19.  There are 0 leftover pips.


Diagram D -- Red:


                     Quads:  Four Q2's is 8, plus 8 (in Q1) = 16. 


                     Squads:  33222 is the "Chair" (Pair + Duck), for 18.  The 5 point combines with two 1-pointers on the mid for two Wides, that's 20.  There are 5 pips left over (the 3 pt blot + two mids).



TOTALS:  Black leads 19:0 to 20:5, for a difference of 1:5, or 11 pips.




Before we determine cube decisions, note that this diabolical diagram contains a couple of advanced pitfalls.  Knowledge of the Special 5-Prime (8 thru 4 pts), could be just enough to get you in trouble.  If you shift the 6-point checkers, one forward and the other backward, you have achieved that prime.


However, under the Squadcount Trouble-Shooting section, a warning is issued (with explanation).  Shifting from 6-5 should be done before the original Quad count, or if afterwards, one needs to remember to adjust by subtracting one Quad.  Deciding never to shift the 6-5 is the safest policy, though that may deny what could otherwise prove to be a useful resource.


"Quad-hopping" can often achieve a desired goal without encumbrances.  Any time you combine checkers from different quadrants to form a Squad, you are essentially employing this technique.  Moving checkers 1 or more exact Quads maintains their parity, while positioning them into recognizable patterns.  In the position 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's position.


Even without knowledge of the Special 5-prime, hopping both 6's down and covering the 5 point with a third clarifies our view.  By banding the entire army of checkers together, all the ways of combining workable Squads, 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; and so on.


Red's position contains a different trap.  Shifting the 1:2 ("8pt) checker forward and the 1:0 ("6pt") checker backward (this time it is the okay direction, because it does not step over a real 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 by them [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 a 5-prime.  We can now more easily count Red's position as a Double Layer plus either two Wides or a Fin; or a Chair and two Wides; either way, with 5 pips left over.


All roads lead to Rome; the preponderance of small counters makes it impossible to oversquad here.  A Quad-hop of the 3-point checker back to create a Triple Layer still leaves a Wide to grab, with a single 5-pip checker as a remainder.  Even if we employ the stinkiest technique possible by using the Sock to mop up all the one-pointers, the [Pair + Fin] or [Pair + Kick] rescues us.  How did you do with this diagram?


Okay, now that we have seen a variety of ways to quickly arrive at the correct totals, we will figure out what to actually do with our count of 19:0 to 20:5.  What are the correct cube actions?




Naccel Race Formula


Truncate the Leader's count to just the Quad number (nothing after the colon).  Now,


                     If 12 Quads or more, subtract 1, divide by 2, and round down.

                     OR, if less than 12 Quads, simply subtract 6.


This is the minimum lead (in pips) necessary for a correct DOUBLE. 

Add 1 to get the minimum redouble, or

Add 4 to get the maximal take point.  [If leader has 18+ Quads, add 5].




That's it.  No percentages.  The Naccel Race Formula matches Robertie's [8 / 9 / 12%] Formula surprisingly well. 


Because the [8 / 9 / 12] Formula is not intended for use below 12 Quads (72 pips), Naccel's simple [subtract 6] rule then approximates Trice's short-race (no wastage) formula, which is [Subtract 5 from PIPcount, divide by 7, round down for take point, subtract from that 3 for redouble or 4 for initial double].


For example, the leader has 9:0, 9:1, 9:2, 9:3, 9:4, or 9:5.  Truncating, and subtracting 6, Naccel gives a doubling point of 3, a redouble of 4, and a take of 7.  Trice's formula agrees with all numbers in this range (54 thru 59 pips).  There are a few spots elsewhere in which Naccel's thresholds differ from Trice's by 1.


It is tricky to find the right balance of close fits and easy implementation, though I am happy with the way the Naccel formula has turned out.  Currently, I am unaware of the existence of any other reference formulas or tables which would allow me to sharpen Robertie's or Trice's approximations.  (Weaver's clever "10% -2 / -1 /+2", though highly practical for traditional pipcounts, does not offer Naccel accurate enough numbers to emulate). 


Please let me know if you find any Naccel examples for which the thresholds stray from these or other known formulas, so I can review them for possible revision.




Let's apply the Naccel Race formula to the count of 19:0 to 20:5, difference of 11 pips, we derived from the most recent position:


            ***  [19 - 1] divided by 2 equals 9.  ***


So, 9 is our minimum double.  10 is our minimum redouble.  14 is our maximal take.

(We added 5 to derive the take point, because the leader's total is at least 18 Quads).


The actual difference is 11 pips.  Black has a solid redouble, and Red has an easy take.




Remembering the Count


All counting methods are vulnerable to a common disaster.  What if you have completed the count for the second player, but have forgotten the count of the first player?  Well, there's not much you can do at that 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 of prevention is better than a pound... on the head.


I heard or read (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 hand to the outside of the board, immediately in front of the point that corresponds to the first digit (or two) of the count, and a finger of your right hand in front of the point which corresponds to the last digit. 


So, 113 would be "bookmarked" with your left finger by the 11 point, and your right finger by the 3 point.  In theory, for 89 you would have your left finger by the 8 point and your right finger on the 9 point, causing your hands to cross (to avoid confusion with 98), though I suppose instead holding one finger at an angle in such cases could clarify that.  Presumably, pipcounts higher than 129 would just ignore the first digit.  So, for 130, you could left-finger the 3 point and right-finger the bear-off tray.


Possible drawbacks to this might be that your arms might tire while you are counting the other color, it might inform an observant opponent of the pipcount (in case that matters), and it just plain looks silly!  However, the method works, and it is easy to remember.


The board could be employed in a similar manner to remember Naccel's counts, but because I was raised to keep my arms and elbows off the table, I recommend an alternate method:




The "Handy" Count


                     Choose a primary location to rest your left hand (could be on your thigh or alongside your chair).  Keep track of the last digit of your Quadcount by extending 0, 1, 2, 3, 4, or 5 fingers.  For a higher digit, move your hand to a secondary location (could be your knee), and extend 1, 2, 3, or 4 fingers to denote 6, 7, 8 or 9 respectively.


                     Extend the fingers of your right hand the appropriate number of Residual Pips.




It is unnecessary to bookmark the first digit of the Quadcount.  For example, 12 Quads is virtually a bearoff position, and it would be very hard to confuse that with the 22 Quad length of, say, the opening position.


For best Naccel results, I recommend use of the Handy Count.  As your speed increases, you may wish to gradually drop the use of your right hand (residual pips), and eventually even your left.


The Handy Count can also be tagged to the Comparison count (see next section).  Your left hand can bookmark the Quad difference (palm up means up Quads, palm down means down Quads), while you figure out the Top-Heavies or Residual Pips.


If you ever see me holding MY fingers up by the board, it will not be a new method.  You will know that I'm either having one of my fits (usually accompanied by slobbering and guttural noises), or I am hoping you'll know how to read the bogus pipcount I will be subliminally signaling.





a[N accel]erated COMPARISON count


                     Step 1:  Compare Quads.

                     Step 2:  Compare either Top-Heavies, or Residual Pips.

                     Step 3:  Add or subtract.




I recommend the traditional method of Comparison counting IF, and only if, you find yourself in a position highly symmetrical to your opponent.  In this case, sizing up a few adjustments will be a little faster than Naccel's method.  In all other positions, I recommend Naccel.


For positions which are difficult to comparison-count using the classical method, Naccel is profoundly practical.  In many cases, you will be able to stop right after the first step -- the "Quad Comparison".   While a close count is not guaranteed, accuracy to the nearest Quad is typical.


What this means is that you can often garner sufficient information during Step 1.  If you are cubed, and find yourself down several quadrants in a light contact position, you are NOT going to take the double -- it is merely a waste of time to figure out the relatively small swing in the remaining pips.  Top-Heavies or (especially) Residual Pips are just "fine tuning".


Similarly, if it is a question of doubling, and a simple Quad Comparison reveals you are not ahead, there is no point in sharpening the count (unless the upper halves of your opponent's quadrants look much heavier than your own).  Other counting methods, which lump it all together, offer no such relief.


If you want to know the race to help you decide whether to make an aggressive or a passive checker 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 your privilege, should you choose to exercise it.




Comparing Quads


One approach to comparing Quads is just to count the total Quads for both sides and subtract one from the other, as we have been practicing.  However, if we have decided we are not counting for purposes of applying the race formula (and thus have no need for totals), we can adopt an even shorter procedure, elaborated here.


We do not normally compare "Speedboards" by counting their pips, but imagine how easy it would be.  You could just go from point 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, so now I'm only down 4 pips.  She has three extra on the ace point, so altogether I'm only down a pip."


Comparing Quads is an identical procedure.  While not as vertically compact, the opposite-colored checkers are on each "big point" are right across from each other, just as in a speedboard, so it is easy to see the differences.


Additional cancellations can help, as long as one does not have to strain to find them.  If Black has a checker on the roof (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 4 point of a speedboard, while Red has a couple extra checkers on the 2 point).  Other common cancellation possibilities include 433 vs 22222 (five on the mid),  Or 33 (an anchor) equals 222 (three on the mid).


These supplemental cancellations can be quite useful in asymmetrical positions.  In symmetrical ones, they will tend to increase the likelihood of a "ghosting" error, and are not needed anyway.  Unless a peripheral cancellation is a clear and easy gain, it is more practical to stick to the straight quadrant-by-quadrant comparison.


For purposes of a Quad comparison, it is often convenient to count a 5 or 4 point checker as a Quad.  Good examples of this are 666 vs 665, or 6666 vs 6664 (or "24 24" vs "24 23").  Pre-canceling such groups is easy visually, and will likely improve your estimate (4 or 5 pips is closer to 6 than it is to zero).  Even if you plan to do a full Comparison (with residual pips), you can carry around a pip or two for later adjustment -- you will save not having had to count the Quad one way, and later the 4 or 5 pips the other way).


If such a cancellation does not feel clean or "fair", it may be because you are seeing other 5's and 4's which seem to warrant weight too.  That is a signal you should go ahead and perform the standard Quad comparison, and then hone it by applying a more comprehensive 5-4 adjustment, described in the next section.




Comparing Top-Heavies


This optional 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 all quadrants) each side has.  The idea is to correct the Quad-count, in less time than it takes to perform the totally accurate Residual Pipcount.


The Top-Heavy Adjustment is a trade-off between time and accuracy.  It seems warranted when the combination of (a) and (b) below seems compelling:


                     (a)  I notice some 5th and/or 4th point checkers sitting out there.


                     (b)  The position is of the type that I don't need an exact count, but in case my current count is off a Quad or  so, it is somewhat likely that I will make an error.


Theoretically, one would like to count 5/6 Quads for each excess 5, and 2/3 Quads for each excess 4, but that is way too complicated.  Even treating 5's differently from 4's seems like more trouble than it is worth.  Although you can tailor your particular method of adjustment to the level of accuracy you wish to achieve, I recommend:




TOP-HEAVY ADJUSTMENT:  Adjust 1 Quad for each 5 or 4 one color has in excess.  If the difference is 3 or more, Subtract 1 from this adjustment.  In the unlikely event the difference is 7 or more, subtract 2.




The Top-Heavy Adjustment is an easy, level-headed compromise, if you feel the count needs a bit more accuracy, but not an absolutely pinpoint.  [If you then change your mind and feel, after all, you need the greater refinement of a Residual Pipcount, it is like changing horses in midstream, but still possible:  Just back out the T. H. Adjustment].




CHECKER PLAY based on Quad Comparison


Knowing the race can often help us choose between what I term "Passive vs Aggressive" plays.  ("Safe vs Bold", which considers exposing a blot, as in Diagram G, is a subset).   For that purpose, I find that to think 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 is usually sufficient to stop after a quick simple Quadcount, without refinement.


Below is an early bear-in position from New Ideas in Backgammon, # 33 (p.97).  Black is considering how to play a roll of 6-4.  Is it better to run around to 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, write down your Quad-comparisons and Top-Heavy Adjustments; then check those posted under the diagram.



          Diagram E

Diagram E -- Comparison:


                     Quads:  Black's back anchor (two Q3's) cancel Red's six outer Q1's.  Another Black Q3 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 has only 2.  Adjust Red's lead from 6, to 8 Quads.




Though a 2 Quad adjustment is unusual, this Top-Heavy adjustment is a slight over-swing of the pendulum (actual count is 7:3, so 7 or 8 was the closest it could come).


So, what about this roll of 6-4?  Is making the outer anchor an aggressive or a passive play?  The answer is "passive", even if we grab the 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 only until next roll, when he starts to realize that what we are actually holding is a water pistol.  Running around to the 1:4 ("10") point is actually the aggressive play, maintaining the back anchor for more real contact.  It is important to get this conceptually straight, so that we know in which direction the race deficit will affect our play.


After we get a Quad Comparison of 6, we can ask ourselves:  Do we feel that the relative race zone we have estimated renders our play of 6-4 a close decision?  If so, might fine-tuning the count affect 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" to sharpen the count is all done in an intuitive wink of the eye).


At a race difference of 8, or 7:3, or even 6 Quads, it is correct to run around (to the 1:4 point) with the 6-4, and by a huge margin.  Kit Woolsey, in his book, offers insightful analysis, and does particularly well to emphasize how the large race deficit 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 race and timing are closely related.  At some reduced relative count, various racing or showdown scenarios will become a practicable option; it will become preferable to create a stepping stone, rather than to keep forces divided solely on the sole merit of the additional shot equity yielded as a result of clinging to the deepest point.


Being informed pedantically that a certain move is "clearly" correct, in no way invalidates 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 to learn how to better evaluate a particular class of positions, and to mentally adjust our relationship between certain race parameters and play thresholds.


It turns out in this position that one has to shift the race further than I would have guessed to swing the correct play.  If one moves both ace-point checkers and a 6-point checker back to Red's midpoint, Black's race deficit has been cut from 7+ to 2+ Quads.  Only then does it become correct for Black to partially abandon the back anchor and make the flexible outside (2:4) point.


For some cube or checker play decisions (though not in this example), you may wish to acquire a count to the exact pip.  To this end, you will be soon be shown tricks for performing quick Residual Pipcounts.  An entertaining one (not necessarily fastest) for the above position could be:


"Remove a Black 0:2 and a Red 1:2 checker.  Stack Black's 4pt checkers onto the 3pt, and demote his back anchor one point to compensate.  This sets up a horizontal symmetry vanishing all 14 checkers which remain on the far side of the board.  Now, Red's midpoint is worth one of Black's 3 pt checkers, and the other three make 1 (s)quad, with 3 pips left over, for a total of 7:3"


This illustrates 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 by blow, after you have read the Residual Pip section.




CUBE DECISIONS based on Quad Comparison


We have seen how the Quad Comparison can affect what kind of a checker play we might select.  Now let's look at the other reason for comparing Quads:  To decide 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 is a worthwhile book, in spite of the fact that I am in there as one of Jerry's victims).


Compare your Quads, and then if you think a Top-Heavy Adjustment is a good idea, you can practice that too.  Write down your answers, and then check to see if you arrived at the correct count.



          Diagram F

Diagram F -- Comparison:


                     Quads:  Cancel Red's three Q3's with 9 Black Q1's -- his extra 6 point checker, and the 8 on the outside.  Red's 2:5 ("17pt") checker offsets Red's two Q1 checkers.  This leaves only Red's two midpoint checkers, which count for a 4 Quad deficit.


                     Top-Heavies:  Black has 4 Top-Heavies, but Red has 5 (one extra).  Adjust Black's Quad lead to 5.




I noticed Red's checkers on the 5 point, and then some high points in the outfield.  In addition, I felt a rough Quad count was insufficient information.  A 4-Quad lead here made this position a solid take in my mind, but more Quads, I wasn't so sure; and less quads, then I wasn't sure about the double.  The presence of high-pointers and cube sensitivity clearly indicated the necessity of a Top-Heavy adjustment.


Knowledge that Black's lead is actually around FIVE Quads is enough to give me complete confidence I should double, and redouble.  It is even enough to lean me towards passing.  The few immediate possibilities of hitting Black are ameliorated by the two blots in Red's board, and the holding game equity appears insufficient at that big a racing deficit.


In the actual game, Black did not even proffer an initial double.  Perhaps he was operating on a certain general principle which advises, when bearing in against a semi-primed holding game, to get the straggler home and lose the market small.  I have to wonder, though, if Black counted 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 for later, yes, but, more crucially, for any immediate hits.  However, in these key variations, she also hits less often (198 vs 296 in 1296), so all is largely offset.  The main factor of moving Red's checkers forward is a gain in the race, a guarantee to be gammoned on the run less often, and to win the race more often.


This alteration slices Black's racing lead from 5 Quads to 3, turning a questionable take into what is not even an initial double.  Such a sizeable swing in cube strength illustrates how important it can be to refine the straight Quadcount when one seems to be in an uncertain range.  And, of all possible contact positions, a simple holding game, which this diagram is rapidly approaching, is the main candidate for cube sensitivity based upon race.


Thus, it is quite conceivable that one might feel a Residual Pipcount is in order -- a tie-breaker to decide a close pass.  This time, the RP count below uses squad comparison, which you can probably already follow.  Anyway, you will have a chance to peruse the myriad of Residual Pip cancellation options in the upcoming section.


Residual Pips:  Black's "Force" near the midpoints cancels Red's "Double-Wide" there.  Black's Duck, Wide (from 2:5 and bar point), and inner Pair cancel Red's Big Triangle (home board).  Finally, Black's remaining bar pt blot cancels Red's ace pt blot, leaving Red with three checkers 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 Quad comparison, a bad miss.  In recognizing the 5 and 4 points looked heavy, we opted for a quick Top-Heavy adjustment, which caught 6 of them.  Performing a Residual Pipcount instead, though a longer procedure, would have caught that 6 plus 3 more.  As "luck" would have it, these 3 extra pips turn out to be enough to nudge this holding game past the take point.




Comparing Residual Pips


There are two main reasons that you might wish to compare Residual Pips:


(A)  You feel that an exact comparison count of a contact position might actually swing a checker play or cube decision.


(B)  Your intention had been to spot-check a cube 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 Total counting because it is faster.  You now realize that a Residual Pip comparison may confirm the race to be close enough to Quadcount the leader, and apply the race formula.


Basically, speedy Residual Pip comparison comes down to the ability to quickly recognize cancellation possibilities.  Here are some options:


                     (1)  You can cancel Squads on one side with similar or different Squads on the other side.  As you are ridding these for both colors at 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, if possible.


                     (2)   Checkers on the Acepoint, Barpoint, Midpoint, and/or Roof ("4:1") are called "Aceys".  Counting one pip each, Aceys are very flexible for offsetting, and not just each other.  Two of them will cancel a checker an x: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, or counter a minus pip-shift.


                     (3)   Any checker in VERTICAL opposition to an opponent checker can be offset.  For example, if both sides have a blot on the 4 point, or both own the 2:5 point, these cancel.  (This is the only type of symmetry which also works in classical comparison counting).


                     (4)   Checkers can be offset HORIZONTALLY, symmetrically around the bar.  Black's 3:2 ("20pt") anchor will cancel against Red's 1:2 ("8pt").  Or 3:4 ("22pt") offsets opp's 1:4 ("10pt").  A Black ace point checker offsets a Red midpoint checker.


                     (5)   Checkers can be offset using INNER symmetry, within a quadrant.  A Black 3:4 ("22pt") anchor balances Red's 4 point right next door, so once again all four checkers disappear.  Applying the same principle, Black's bar point offsets Red's midpoint -- the opposite end of the same quadrant.


                     (6)   The final symmetry possibility is "HOP-SYM", found by hopping a checker 6 pips and then applying either horizontal or vertical symmetry.  Black's 4:0 ("24") point offsets Red's 3:0 ("18") point.  Or Black's 3:4 ("22pt") offsets Red's 2:4 ("16pt").  This symmetry is the hardest to notice, but can sometimes prove handy.


                     (7)   If a desired symmetry doesn't match up exactly, you can force it, then mentally SHIFT checkers elsewhere the same number of pips in the opposite direction (or just add or subtract a pip or two from a running count).  Or, you can (for any reason) swap ANY checker, point, or group, with that of the opposite color -- ANYWHERE on the board!


Note that "Quad-hopping" is a combining technique for same-colored checkers, not a cancelling technique for opposite-colored checkers.  You can, of course, Quad-hop to position a Black checker, as long as it is symmetry which causes the actual cancellation (see "Hop-sym" above).


Keep on the lookout 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 the point that within a few seconds you will just "see" the Residual Pip difference.




Fine-Tuning Checker Plays


Decisions whether to expose a blot or not (Safe vs Bold play) are generally more sensitive to the exact pipcount than are some of the less volatile types of Aggressive vs Passive plays (such as the 6-4 play in Diagram E).  A small race adjustment can swing a play more easily.


With this in mind, 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 have learned.


We are Black, 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 you guess it becomes correct to switch plays?


For each position 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, but not both) of the Top-Heavy or Residual Pip alternatives.  However, for purposes of teaching, I am asking you to practice both full comparison alternatives (Quadcount + Top-Heavy, and Quadcount + Residual Pips), and compare your answers with those found below.



          Diagram G

Diagram G -- Comparison:


                     Quads:  Black has an extra checker in Q3, while Red has his extra in Q1 ("6pt").  Red has a 2-Quad lead.


                     Top-Heavies:  Black has 3 Top-Heavies, Red has two.  Adjust Red's Quad lead to 3.   OR:


                     Residual Pips:  Swap Red's 5 point with Black's 3:3 next door.  Horizontally cancel one of the Black's new 3:2 checkers with a Red 1:2 ("8pt") checker.  This leaves Black only with his 2:4 checker, counting 4 pips.  Added to the 2 Quads, this makes a race deficit of 2:4.




In the Safe Play vs Bold Play chapter of Paul Magriel's book (which is still often referred to as the "Bible"), the above position is given, with the example of a lone 3 roll accompanying the diagram.  The recommendation was to hit, but that is a substantial error, as Bagai points out 25 years later, because doing so breaks the 1:2 ("8") point. 


Hitting with the 3 reflects the perceptions and priorities of the time.  Strong player's of the 1970's battled over key points aggressively, with little concern for the race.  There was an unspoken agreement that the only skillful win was one in which a pure position triumphed over a cracked one -- and especially deserved if one first maneuvered into a backgame or beautifully complex holding game, well down in the race.  It was a charming era, to be sure.


I know; I learned in that environment.  I even unwittingly helped perpetuate the myths.  They/we played these poor positions quite well, and even got away with it handsomely, until the level of play improved.  Eventually, opponents no longer buried early when they could slot, or took crushing redoubles.  Nor did they go to the other extreme of dangling so many carrots that it became hilariously profitable to abandon a backgame, or pass bluff re-whips.


Fortunately, we can still pay homage to this biblical archive by altering the roll to 3-2.  I will assume there are two 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 four blots, though it is an "opportunity" at which many '70's players might have leapt).


Perhaps there are players who would hit with the 3-2 even if way ahead, and others who would play safe even from way behind.  For those players, counting would be irrelevant.  But most players would probably have some race threshold (either exact or rough-range) that would help decide the hit.  Consulting the race is likely to help. 


It turns out that in this position, because we are down 2+ Quads, it is very right to hit with the 3-2.  If we reduce 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 the midpoint, the margin narrows by about 1% per pip pair (there is a safe-warp at the 16 point, but that has to do with the value of using the 2 to connect the back checker), and by the time we get to the 14 point, it is nearly a tossup.  With the aid of rollouts, we have discovered the threshold:  We should hit if down 2 pips or more.


If we further 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 you will guess by now, playing safe is (very) correct even though we can hit without breaking the 1:2 ("8") point.  It is a pity that Paul did not either choose our 3-2 roll, OR stick with the lone 3 but add a builder to the 1:2 point (from the mid); either way, that diagram pair would have supported his well-conceived theme perfectly.  It is also a pity that Paul is such an honest fellow; otherwise he could claim his diagrams were misprints.


Contrary to popular belief, the issue of having more versus fewer checkers "back" plays second fiddle to the race as a  criterion for choosing a Safe or Bold play.  Having established the 2:2 ("14") point as a Safe/Bold threshold:  If, from there, we move two checkers from Red's mid to her "6" point (to refund the 14 pips we stole, in the way which least affects immediate tactics), hitting is correct by 7%, just as in the original diagram.  In spite of the fact that Red is now the one with the extra checker "back", we make the bold play because, as before, we are down 2+ Quads.


That we might not know that the Safe/Bold threshold is -2 pips (or even in the neighborhood of zero Quads), is no excuse for throwing up our hands and not counting the race; a better informed decision will be right more often.  A more rational excuse is that we are slow at counting, and don't feel it is fair to keep our opponent (and perhaps by ripple effect, half the people in the tournament hall) waiting.  I applaud that "excuse"; hearing it is like a breath of fresh air.  But don't let me get started on that.  It suffices to say that if we get faster at counting, we no longer need to be a slave to that excuse, nor to chugging away at tedious arithmetic.


Quadcounts are off about 4 pips on average (though can stray a fair bit further).  Will this inaccuracy make a difference in the play that we select?  Sometimes. 


If we feel that our decision is race-sensitive and the Quadcount is on the bubble, then we can add in the Top-Heavy Adjustment, which is quick and cuts the inaccuracy in half.  Or, if a Safe vs Bold Play position is close to a reference position we've seen, and/or we are capable of making sound adjustments, then the pinpoint accuracy of a Residual Pip comparison could well determine our choice of play.




Racing Onward


Don't worry!  You are not expected to remember all of the counting nuances in this article.  If you have even followed, let alone absorbed, ten percent, you are doing very well.  The numerous options are not meant to overwhelm you, but to be your friend, so add what you like, a little at a time.  Meanwhile, Naccel can function quite adequately on the mere fraction you have digested.


One of the major 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 brains over time) which indicate exact (or rough-range) thresholds at which we should alter certain checker plays or cube actions.  While the race is only one contributing factor in making these decisions, it is a more dominating one than players in the past have realized.


Backgammon is, in essence, a race.  The respect modern players have gained for the race is reflected in the types of positions for which they strive, and in their inclination to count a greater variety of positions.  For this purpose, it makes 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 count the race far more often, while investing less overall time doing so.




If you would like to continue practicing your counts, verification of accuracy is much easier if pipcounts are pre-posted next to the diagrams (to convert, multiply your Quads by 6 and add the residual pips).  Suggested sources are:  The Woolsey and Bagai (only 3 count errors each -- can you find them?), and Ortega/Kleinman books, from which this article borrowed diagrams, and Mario Kuhl'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 a button.


Our deep appreciation goes to Ric Gerace for providing such legible color diagrams, and incorporating so precisely the new quadrant numbers, point numbers, and quadrant divisions.  (You can find some cool stuff on Ric's website, which is


This is the full-length August 2001 version of the Naccel article.  A slightly updated, though edited, version will appear in September in Backgammon Today.


Ulf Wostner is planning to construct a website, for the purpose of teaching Naccel.  If successful, this article will be reprinted there, in expanded form, along with reader input and various teaching mechanisms, gradually improved over time.  The address is, though we do not know how soon it will be functioning.