The vast majority of the time that you are playing a backgammon game, it is unnecessary to know the total or absolute pipcounts; for example, that you are leading 106 to 132. (Either it is not a straight race or the lead percentage is not in a range that compels a race formula.) It is quite sufficient to know the difference is 26 pips or even just about 26. The determination of this difference is called a comparative or relative pipcount.
To this end, "colorless counting" can be considered a shortcut. It is based on the elegant axiom that any checker on the board can be swapped with any other checker on the board, even one of the opposite color, without affecting the relative pipcount.
Logically, then, one can IGNORE the color of the checkers entirely -- just treat all thirty checkers as if they are the SAME color.* This concept has been around for several decades, but it is only recently that colorless-counting systems have been formally created.
* The "exception" is a checker on the bar, a peculiarity of backgammon placement that is no fault of any pip-count system! A checker on the bar is actually on the player's imaginary 25pt or the opponent's imaginary ZERO point, which is essentially the same as the opponent's bear-off tray. Thus, imagine any Blue checker on the bar in the White bear-off tray. Conversely, imagine any white checker on the bar in the Blue bear-off tray. Once there, you can ignore the color, just like any other checker.
Colorless counting allows us to devise shortcuts that do not exist when we are constrained by having to differentiate color. One bountiful example, which I've discussed before, is vertical cancellation. Granted, this concept exists in color-restrictive counting as well: you can cancel a blue checker on (say) Blue's 5pt with a white checker on White's 5pt without affecting the relative pipcount. The difference is that in colorless counting, you have not only those opposite-color cancellations available but also same-color cancellations (blue vs blue, and white vs white).
I'll now introduce quasi-vertical cancellation. In the simplest form, the checkers form a symmetrical trapezoid.
Forgive me if you've already hardwired in the concept, but for the sake of other readers I'll be interspersing reminders that checker color makes NO difference. In the diagram below, the four checkers could just as well be all blue, all white, three blue and one white, or whatever.
[While checker color doesn't matter in colorless counting, it is still necessary to define or designate which left/right side the checkers are bearing off, and/or which player is at the top or the bottom of the diagram. In my diagrams, the Blue player (capitalized "Blue") is always at the bottom (the "near side"), and the White player ("White") is always at the top (the "far side").]
White's
Bear-off Tray
Blue's
Bear-off Tray
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Trapezoid | |||||
One way to describe this formation is that it is a rectangle (two pairs of vertically cancellable checkers) with the wrinkle that the checkers on either the top or bottom are one pip "off" in each direction.
You can, if you like, create a rectangle by (say) moving the 2pt checker to the 3pt, and the 11pt checker to the 10pt. This utilizes offset-shifting Law #1: Checkers on the same side of the board can be moved equally towards or away from each other without affecting the count.
Instead, I recommend that you embrace the diagrammed formation as a "trapezoid" if you can, and not bother with those one-pip movements. The distinction may seem subtle, because one-pip shifts can be visualized quickly, but instananeous is better than quick! (The superiority manifests in more than just the small savings in time; there is also a reduction of disturbance and fallibility to your train of thought.)
A trapezoid can be as narrow as wide as the board allows. For example, try moving the bottom checkers in the other direction -- away from each other, to the 1pt and 12pt. That gives you a slightly wider trapezoid, which again is an instant cancellation (i.e., all four checkers disappear -- poof, with no effect on the count).
Leaving the bottom checkers either in their original locations (2pt and 11pt) or in the corners (2pt and 12pt), experiment now by moving the top checkers gradually towards each other, and you'll form progressively narrower trapezoids.
[I use "wide" and "narrow" to compare the sums of the checker-to-checker distances at the top and bottom. However, as the angles widen at the top, they narrow at the bottom, and vice versa. If you somehow find the terms wide/wider and narrow/narrower counterintuitive, disregard such references. What is important is that you recognize the symmetry, which (with practice) will cause checkers to vanish more quickly from the board.]
Checkers in a trapezoid (or a rectangle, for that matter) do not have to be equidistant from the raised bar in the center of the board. Another way to adjust a trapezoid is to move one top checker and one bottom checker in the same direction. This utilizes offset-shifting Law #2: Checkers on the opposite side of the board can be moved equally in the same left/right direction without affecting the count.
For example, if the 2pt checker and 15pt checker are both moved to the left, you have a different (though equally "wide") trapezoid. In a similar vein, if the 2pt checker and 22pt checker are moved to the left, you have a different (wider) trapezoid.
[My purpose in encouraging you to mentally move checkers in this segment has been to elucidate different trapezoids (without taxing this post with a bunch of slightly different diagrams) in order that you can gain familiarity with their appearance. Your goal is to be able (during counting) to spot trapezoids that already exist on the board (without having to shift them to rectangles). You can "cancel" (poof) any such symmetrical trapezoid formation, just as you can with a rectangle.]
Can you can spot the same trapezoid (that is diagrammed above) from the more complex arrangement below?
White's
Bear-off Tray
Blue's
Bear-off Tray
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Vertical cancellations, and Trapezoid | |||||
Did you find the trapezoid? On the right, the two blots are the same as in the previous diagram. On the left, what were formerly two blots (on the 2pt and 22pt) are now two SPARES. These four checkers form an equally valid trapezoid.
Mentally remove the trapezoid checkers (comprised of the two blots and two spares) from the board. Can you picture what remains?
What remains is a prime with a hole on the 4pt, on each side of the board. (Remember, checker color is irrelevant.) The primes, being completely symmetrical, cancel out. The entire count of this diagram is ZERO.
In practice, you may find it easier to do the straight-vertical cancelling first, then cancel the trapezoid. ("Oh, cool, I see what I have left is a trapezoid.") You can even do both at once (if you are able). It may be a matter of taste or depend on the circumstances.
Okay, good. Let's put trapezoids aside for now.
There is a lucrative technique called "hopping." It can be thought of as one-way shifting on a large scale. Each "hop" is equivalent to one UNIT of counting, whatever that unit might be within a particular pipcount system. (I first introduced "hopping" when explaining Naccel, a system that counts in units of 6 pips.)
Typically, you "hop" a checker or checkers for the purpose of forming a group (the larger the better) for which you recognize an instant count. For example, if a single hop creates a group that counts 4 (units), a combined count of either 3 or 5 results (depending on the direction of the hop).
Twip is a system that counts in units of 12 pips. You can, for example, hop a checker from the 22pt to the 10pt, or from the 2pt back to the 14pt, for a 1-twip adjustment to the count. However, there are other -- more common -- ways, to hop.
The more useful Twip hops tend to be with two checkers (6 pips each) or three checkers (4 pips each) or four checkers (3 pips each). From the single diagram below, I'll demonstrate the three-checker kind (i.e., hopping three checkers, 4 pips each).
I should insert a couple of preludes. First, the standard reminder: Checker color is irrelevant. The checkers on either or both points below can just as well be blue or white, or even mixed colors on the same point!
Second, while I refer to the points by their traditional point numbers (because that's what you're used to), it is the shared "mid tray" (midpoint tray) on the right side of the board that is conceptualized as the ZERO point in Twip. The "bear-off" trays, to the left of the home boards, are counted as –1 or +1 twip (being 12 pips away from the Twip-system's mid tray / zero point on the far right). In time, thinking of these as counts of "0" and "1" will be second nature (and really fast!).
[Technically, the ZERO point is the half-pip ridge between the mid tray and the midpoint. Put another way, the mid tray and midpoint are a half-pip on either side. Hence, to make counts quick and easy, count the mid tray as zero, and count the midpoint as zero. The half-pip slop is well worth it (and when you occasionally want exactitude, it is easy to correct for it).]
Likewise, on the left side of the board, count the bear-off tray or the 1pt as –1 (twip) on the far side, and as +1 on the near side.
White's
Bear-off Tray = –1
Blue's
Bear-off Tray = +1
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A good situation to HOP | |||||
In the diagram above, moving the stack of three far-side checkers 4 pips to the right (i.e., 20/16(3)) creates a symmetry, a vertical cancellation. This 4-pip movement of three checkers is a 'hop" of 1 twip. As this hop moves away from the opponent's bear-off tray, it counts –1.
Alternatively, you can "hop" the near-side checkers to the left with 9/5(3), and, by the same token, the hop movement counts –1. (Either way, pocket the count of –1 while the six checkers vanish.)
[Plusses and minuses can be confusing in ANY pipcount system (made all the more so because people don't apply them the same way). The standard I use for comparative counts is that plus (+) is favorable for Blue (represents/shows that he is leading), and minus (–) is unfavorable for Blue (shows that he is trailing).]
If you have to move checkers (regardless of color) towards Blue's bear-off tray, or away from White's bear-off tray, in order to make the position equal/symmetric, that demonstrates that Blue was trailing before you made the (mental) move. Therefore, in this diagram, the equalizing "hop" of 20/16(3) or 9/5(3) is counted as –1 twip (i.e., –12 pips), because Blue is shown to be trailing by that amount.
Still viewing the same diagram above, let's try (though clearly less efficient from a global perspective) counting the far-side and near-side checkers separately:
If we move the far-side checkers to the left, it is a hop of +1, and the three checkers on the 24pt count as –3 twips (i.e., –1 each), for a net of -2. Alternatively (starting again from the 20pt), we could hop this three-stack twice to the right and "bear them off" to the mid tray, each hop counting –1. That's another way to produce (or verify) the count of –2.
On the near side, suppose we hop those three checkers to the right, into the mid tray (where they count zero). That hop is a count of +1. (Another way to think of it is that when we back checkers away from Blue's bear-off tray, we have to compensate him; hence the adjustment is plus.)
Let's combine the counts in the above two paragraphs. The far-side count is –2, and the near-side count is +1, for a net of –1. Great. Naturally, though, our original one-step method was a lot faster, when we hopped (–1) one of the three-checker stacks in the direction that vertically aligns it with the other.
As pointed out earlier, "hopping" is one-way shifting on a large scale. You shift and adjust the UNIT count, instead of shifting and adjusting one or more baby pips. Hopping and other forms of unit-counting typically do not (though sometimes do) completely replace single-pip counting, but in any case they save a lot of time.
More often than not (whatever unit-system you use), there will be one or more baby pips left over, which you can either ignore/discard if it looks like the difference won't affect your cube/checker decision, or you can count them through the use of small-scale shifting. (Or even the old-fashioned way -- shudder -- each checker by point number.)
Let's quickly review the basics of one-way shifting on a small scale. If you move a checker (of either color) 1 pip away from Blue's bear-off tray or towards White's bear-off tray (actually, if you are doing one, you are automatically also doing the other), say from Blue's 7pt to 8pt, the point number increases by one and Blue should be credited 1 pip in the comparative count. In other words, count it as +1 (pip).
Conversely, if you move a checker 1 pip away from White's bear-off tray or towards Blue's bear-off tray, say from Blue's 21pt to 20pt, the point number decreases by one and Blue should be debited 1 pip in the comparative count -- that is, –1 (pip).
Now, suppose we want to use one-way shifting to help count the diagram below.
White's
Bear-off Tray
Blue's
Bear-off Tray
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Shift to achieve symmetry | |||||
You can use small-scale shifting to construct or compile a larger formation. On the near side, you can mentally move 2 pips twice with 5/4(2) and 2/1(2), which creates a global group with both sides' 4pt, 3pt and 1pt occupied. This 12-checker formation can be vertically cancelled, and therefore counts zero. The shift itself counts –4 pips, which is the entire count of the diagram.
Alternatively, you have the far-side shift of 24/23(2) 21/20(2). Again, the count is –4 pips.
There are two other ways of reaching a 12-checker symmetry. One is 5/4(2) 24/23(2), and the other is 21/20(2) 2/1(2). Just be sure you understand that (in both these cases) all four checkers are moving in the same direction -- "towards" Blue's bear-off tray. If you think you might mistake such a shift for a two-checker vs two-checker offset (which yields a zero-pip change) and correspondingly fail to adjust the count, then either restrict yourself to same-side shifts... OR overcome that misperception by reviewing "Law 1" and "Law 2" of offset shifting (written earlier in red typeface) -- which is a good idea anyway.
Now let's apply our counting techniques to a Sax-KG position that Paul Weaver recently posted (diagrammed below).
Prior to vertical cancellation, this position can be counted in three steps, in whatever order you like. I will demonstrate the count in this order:
.....1. Trapezoid
.....2. Hop (a three-checker stack)
.....3. Shift (4 pips)
White's
Bear-off Tray = –1
Blue's
Bear-off Tray = +1
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Remove the Trapezoid | |||||
1. TRAPEZOID:.. Mentally remove the trapezoid, consisting of the two blots and the two leftmost spares. Verify your visualization with the diagram below.
White's
Bear-off Tray = –1
Blue's
Bear-off Tray = +1
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Hop the three-stack out to the 16pt | |||||
2. HOP: ..Hop the three-checker stack 20/16(3). This counts –1 twip.
Once you've "done" this hop, verify your visualization with the diagram below.
White's
Bear-off Tray = –1
Blue's
Bear-off Tray = +1
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Shift -4 pips | |||||
3. SHIFT: ..Shift 5/4(2) and 2/1(2). This counts –4 pips.
When you've mentally completed that shift, verify your visualization with the diagram below.
White's
Bear-off Tray = –1
Blue's
Bear-off Tray = +1
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Vertical symmetry | |||||
This diagram clearly has a count of zero. You reached it by removing the trapezoid (0), hopping the 20pt out (–1 twip), and shifting two low points (–4 pips). Thus, in the original position, Blue is behind by a "twip and 4" (which is 16 pips, though this conversion is unnecessary).
You can transpose the steps and (for example) do the trapezoid removal third. That step can even be bypassed altogether if you are confident enough in your visualization and logic to incorporate the trapezoid into the final symmetry. If so, then arguably the entire count of the original position takes two steps.
I didn't want to clutter up the diagrams, but in the Sax-KG position (the one Paul posted and we've been counting), Blue leads 4–3 in a 9-point match, owns a 2-cube and has a roll of 63 to play.
I proposed a variant in which the only change is that White owns his 2pt instead of his 6pt, which is diagrammed below (again without the clutter of match, cube or dice information). How might you count this position?
White's
Bear-off Tray = –1
Blue's
Bear-off Tray = +1
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Do an offsetting shift: 21/19(2) vs 20/24 | |||||
There are many ways to count. Try this one:
...(a) Remove the trapezoid (the two blots and two leftmost checkers).
...(b) Move 21/19(2) and offset it with 20/24.
Neither of these steps affect the relative count.
I challenge you now to visualize BOTH steps ("a" and "b") before verifying your visualization with the diagram below. Can you do it?
White's
Bear-off Tray = –1
Blue's
Bear-off Tray = +1
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Double-hop with 9/5/1(3) | |||||
Now, "hop" the three-checker stack on the 9pt twice -- 9/5/1(3) -- to reach the 1pt, for a count of –2. That is your entire count!
[OR hop the 24pt checkers twice out to the 16pt for –2. OR count the checkers on the 24pt as –3 (i.e., –1 each), and hop the 16pt stack rightward to the mid tray for +1, which nets to –2.]
I trust you can see that with the two three-stacks aligned (or gone), what remains is a large-scale though straightforward vertical cancellation. The count stands: Blue trails by 2 twips (which is 24 pips, though conversion is unnecessary).
[It should be noted that just because you are counting in Twip, doesn't mean you have to confine yourself to hops of exactly 12 pips. For example, going back to the second-to-last diagram, another solution is to remove the trapezoid and play 24/19(2), 21/16(2) and 20/16, three (negative-count) movements that sum to –24 pips.]
In short, Blue trails by 16 pips in the original position, and he trails by 24 pips in the variant position. Regarding these positions as checker-play problems (Blue to play 63), the difference between –16 and –24 is enough for Blue to switch to a purer checker play. (Admittedly, the fact that White's 6pt is open instead of his 2pt impacts tactical checker-play tradeoffs more than does a "neutral" 8-pip alteration -- if it exists.)
To see rollout results for both positions, click on this post and look for something like "Variant -- Rollout" in the subject header of a nearby post that will almost certainly exist by the time you finish reading this sentence.
Nack