To fully make use of the references we learned in parts 1 through 3 of this series on cluster counting, we will need to make use of the mental shift method we learned about previously.

### 9. Bear-off positions

When bearing off, you very often get to use the 10-checker references with fancy names (*e.g.,* the shed, the triangle, *etc*).

Consider the position below:

Here, you have 10 checkers, but they aren’t arranged in any obvious cluster. *However,* with 2 simple mental shifts (moving the checkers from the 4 and 6 points to the 2 and 3 points) we can rebuild a 5-prime. The center of this mental prime is on the 4-pt, so we start the count with 40 pips. To shift back to the position above from a perfect double-decker prime, we need to move a checker from the 3-pt to the 4-pt, away from the bear-off tray, adding 1 pip to our count. The second checker needs to relocate from the 2-pt to the 6-pt, for 4 more pips. Together, these add up to 45 pips.

A second example of a bear-off position is shown below:

This is almost a perfect 1-2-3-4 triangle, but it’s missing a corner. The key point to look for in this cluster is the 3-checker-high part of the triangle, which in this position is on the 3-pt. Therefore, we start with 30 pips. It’s missing 2 checkers on the 2-pt, so we remove 4 pips, resulting in 26 pips.

### 10. Anchors

We learned from our reference that any checker on the golden point (the 20-pt) is worth 20 pips. We can use this information to speed up counting checkers in our opponent’s home board.

Below, we are holding 2 anchors in a 1-3 backgame.

Without considering clusters, we could calculate the pip count the hard way (24 × 2 + 22 × 2 = 92). Instead, **we can start by saying we have 4 checkers on the 20-point (80 pips) and then mentally adjusting them**. 2 of those checkers are shifted by 2 pips (+ 4), making the 22-pt anchor and 2 others are shifted by 4 pips (+ 8). Adding 4 and 8 to 80 brings us to 92 pips in fewer steps with smaller numbers.

This trick also works with an odd number of checkers. Consider the position below, with an advanced anchor and a goalkeeper:

3 checkers in our opponent’s homeboard start us at 60 pips. Next, we need to account for the extra 4 pips to shift the goalie from the 20-pt to the 24-pt, for a total of 64 pips.

### 11. Symmetry

Mental shifting enables another trick for counting pips efficiently by looking for symmetry in the checker distribution. Consider the position below:

The distribution above would arise after throwing a 42 and playing 8/4, 6/4. It doesn’t seem to fit any of the molds from the first 3 parts. You could count these checkers the hard way: (2 × 4) + (4 × 6) + (2 × 8) = 48 pips. Instead, notice that there is mirror symmetry around the 6-pt. You could shift the 2 checkers on the 4-pt back 2 pips, and shift the 2 checkers on the 8-pt forward 2 pips, and then **all 8 checkers would line up** on the 6-pt:

8 × 6 = 48 pips! Same result in way fewer arithmetic operations!

**This symmetry method is flexible enough to work with slightly asymmetric distributions** as well! Consider the position below, where we make the 5-pt instead of the 4-pt:

This distribution *would* be symmetric if we shifted 2 checkers over, for example moving the 8-pt checkers to the 7-pt. So let’s pretend they’re there! In that case, we, again, have 8 checkers on the 6-pt for 48 pips. To return to the original position, 2 checkers need to be shifted away from the bear-off tray, so we add 2 pips, ending up with 50.

Hopefully, these examples demonstrate the power of the cluster counting method. Eventually, you will start recognizing clusters in arbitrary distributions of checkers, really speeding up your counting by orders of magnitude.

Next lesson: Cluster counting 5: examples

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