Criss-Cross Count (CCC).
Fast pip count method. Tiny numbers. Few operations.

by Max Glaezer (maxim.glaezer@gmail.com), Aug 2018, short link to this document: tinyurl.com/crisscrosscount

CCC calculates pip count difference in few operations.
The method tends to operate with single digit numbers most of the time.

Internally CCC combines ideas from colorless pip count and half-crossover pip count methods.

However, the familiarity with those is not required.

To get an impression of how fast and simple CCC is, consider the following examples.

See more examples in the Practice Guide (https://tinyurl.com/cccpractice).

Example 1

Step 1. Calculate Step1 value using the diagram:

Step1 = 4 + 2 - 5 - 1 + 2 * (5 - 2) = 6 

In zones “++” and “--” each checker counts 2. For example, if zone “++”  has 5 and zone “--” has 2, the result for those two zones is (5 - 2) * 2 = 6.

Step 2. From the number of your checkers on top of the board (2) subtract the number of opponent’s checkers on the bottom of the board (3). Add the difference multiplied by 5 to the result of Step1.

Step2 = Step1 + 5 * (2 - 3) = 6 - 5 =  1     // See how to do it fast in Q&A

Step 3. Do unit adjustments.

Step2 value multiplied by 3 would be a precise pip difference if all the checkers were placed symmetrically within their zones.

Here all the checkers that violate symmetry in their zones are circled.
Look at the zone 4pt-6pt, for example. Without the circled checkers, the remaining 4 checkers have a symmetrical formation.
Looks at the zone 19pt-21pt now. Without the circled checkers the remaining 4 checkers have a symmetrical formation.
Look at the zone 1pt-3pt, without that single checker on 1pt the remaining 0 checkers would have a symmetrical empty formation.

Starting from your bear off tray add together all unit adjustments, keeping the running total in your head.

Do not ignore any checkers when doing unit adjustemens.

UA = -1 + 2 - 2 - 2 + 1 + 3 -1 = 0

With experience you may want to notice and cancel all possible non-symmetrical groups. Please be careful at first.

Step 4.

Calculate pip count difference: Diff = Step2 * 3 + UA

Diff = 1 * 3 + 0 = 3

As in XG, positive numbers mean that we are behind. We are 3 pips behind in this position.

Step 5. Enjoy your pip count.

Please notice that all the arithmetic operations for this position were with single digit numbers. This is often the case.

Example 2

Step 1. Calculate Step1 value using the diagram below.

Step1 = 3 + 3 - 6 - 3 - 2 * 2 = -7

In zones “++” and “--” each checker counts 2. Here, we have (0 - 2) * 2 = -4.

Step 2. From the number of your checkers on top of the board (4) subtract the number of opponent’s checkers on the bottom of the board (1). Add the difference multiplied by 5 to the result of Step1.

Step2 = Step1 + 5 * (4 - 1) = -7 + 15 = 8    // See how to do it fast in Q&A

Step 3. Do unit adjustments.

Step2 multiplied by 3 would be a precise pip difference if all the checkers were placed symmetrically within their zones.

Here all the checkers that violate symmetry in their zones are circled.

Look at the zone 16pt-18pt, for example. Without the circled checkers, the remaining 4 checkers have a symmetrical formation.

Looks at the zone 19pt-21pt now. Without the circled 7 checkers the remaining 0 checkers have a symmetrical empty formation.

Starting from your bear off tray add together all unit adjustments, keeping the running total.

UA = 2 - 1 + 1 + 2 - 7 + 1 = -2

With experience you may want to notice and cancel all possible non-symmetrical groups. Please be careful at first.

Step 4.

Calculate pip count difference: Diff = Step2 * 3 + UA

Diff = 8 * 3 - 2 = 22

As in XG, a positive number means that we are behind. We are 22 pips behind in this position.

Advantages of the method

• Few calculations.
• Small numbers. The way the calculations are constructed tends to produce single digit numbers.
• Minimal memory capacity is required to keep running total of half-crossover differences and unit adjustments.
• Practicing unit adjustments helps in calculating absolute pip count using Douglas Zare’s method (http://www.bkgm.com/articles/Zare/HalfCrossoverPipCount.html).
• It is a psychologically rewarding experience to be able to ignore all those nasty checkers on 4pt-6pt where they tend to accumulate.

Video with More Examples

https://tinyurl.com/crisscrossvideo

100+ Practice Positions with Calculations

https://tinyurl.com/cccpractice

Q&A

How do I account for checkers on the bar?

Imagine your checker on the bar to be on your 24pt. It’s unit adjustment, however,  will be +2.

Imagine your opponent’s checker on the bar to be on your 1pt. It’s unit adjustment will be -2.

How do I account for checkers in the bear off tray?

Imagine your checker in the bear off tray to be on your 1pt. It’s unit adjustment, however,  will be -2.

Imagine your opponent’s checker in the bear off tray to be on your 24pt. It’s unit adjustment will be +2.

What if I find it hard to do unit adjustments without making mistakes ?

You could try doing positive unit adjustments first and then sequentially doing negative unit adjustments.
See if that helps.

You can learn some tricks about unit adjustments in “fine tune” section of The Half-Crossover Pip Count and in “unit crossovers” section of Urquhart Colorless Counting (UCC).

Can I skip unit adjustments (step 3) ?

In case one player is far ahead or far behind in the race, the precise calculations might not be necessary.
Sometimes you may skip the unit adjustments step if you feel the total unit adjustment is small.

How to remember the intermediate results?

You may want to use sign language and “remember” positive numbers with your right hand and negative numbers with your left hand. For example, after Step 2 you may remember the number of half crossovers using your hands. Alternatively, you may triple the number of half crossovers before returning to that number after doing unit adjustments. Alternatively, you may do Step 3 first and remember the unit adjustments using your hands. Unit adjustments tend to be smaller numbers with STD=4.5.

Is there any way to to Step 2 fast ?

Yes! There is a visual way to do that and we believe that is what Art Benjamin does (for a slightly different purpose) when he moves his hand over the board and closes one of his eyes. The trick is simple: starting from the place right between two mid-points adjust the position of your hand as a visual divider so that the number of your checkers behind the divider is the same as the number of your opponent’s checkers before the divider. The number of checkers between the divider and the place between mid-points is exactly the difference required in step 2.

How does CCC work ?

CCC essentially calculates the running total of half-crossover differences on a colorless position.

Zones “+”, “++”, “-” and “--” are simplified representations of zones -1, 1 and 2 in the half-crossover method.

The origins of Step 2 might not be very apparent immediately, but in short:

1. The sum of the Douglas Zare's zone numbers on the opposite sides of the board is 5 (the multiplier in Step 2). For example, 7pt-9pt - “zone 1” and 16-18pt - “zone 4”, and the sum of their numbers is 4+1=5.

2. Subtracting the number of opponent's checkers on our side of the board from the number of our checkers on his side of the board is a neat way to get the number of checkers in excess of 15 on one side of the board (“extra” checkers). Those “extra” checkers belong to “zone 3” and higher. Multiplying their number by 5 effectively adds the right values for those “extra” checkers in terms of half-crossover method for “zone 3” and higher in addition to what has already been done in Step 1.

For example, consider the extra checker is in “zone 3” on our side of the board as if it belongs colorless-ly to our opponent, On Step 1 is is handled with 2x multiplier. And on Step 2 it is handled with -5x multiplied. The effective multiplier for this checker becomes -3x, which corresponds to the zone number. The sign “-” signifies that it “belongs” to our opponent in the colorless position.

Please also read Nack Ballard’s explanation which covers the nature of this method in depth.

The way the calculations are constructed minimizes the produced numbers and possible mistakes.
The unit adjustments step is directly borrowed from UCC with alternative visualization technique.

What is the difference between UCC and CCC ?

Step 1 and Step 2 are a complete replacement of UCC first two steps based on ideas from half-crossover pip count. We believe the suggested calculations are faster and less error-prone. Step 3 is essentially the same, with some visualization technique (symmetry in the zone) that makes it faster.

Speed-wise how does CCC compare with other methods ?

The closest competitor to CCC is colorless cluster count.

Players who are into cluster counting might do the following “colorless” trick for the position below:
they assume that the 15 checkers on 1pt-11pt are black and the rest 15 are white (see diagram on the right).
Now black has (8 + 4) * 5 + 2 (cluster 4pt-8pt) + 1 + 40 + 1 = 104,
white has (8 + 4) * 2 + 3 (cluster 4pt-8pt) + 1 + 3 + 30 + 1 + 3 = 101.
The pip count difference is 104 - 101 = 3.
Pretty fast, if you are into cluster counting, remember several reference positions, arithmetically fluent and deal with clusters and mental shifts efficiently and without mistakes.

In comparison, CCC method deals with much smaller numbers: [4 + 2 - 5 - 1 + 6 - 5] * 3 + 0 = 3.
Some minor math (Unit Adjustments) is hidden behind the number 0 in this example.