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senary spinoffs

Posted By: Don Thompson
Date: Tuesday, 19 January 2010, at 2:40 a.m.

In Response To: Basic squads explained with proper diagrams (Nack Ballard)

I've heard very good things about this website for a long time but never really got enough momentum to get here. Thanks for the nudge to get involved.

I have been using Naccel as my only counting system for a while. Itfs actually the only count Ifve gotten proficient at. It is very fast and efficient especially when you get into the poofs and cross board shortcuts. Once you get used to it, there is really no need to convert back to a conventional count. It can be easier to use converted race formulas except of course if you're in a chouette and don't really want to explain to your partners what 9:3 to 11:0 means ( I've always used a colon as a delimiter if there is a better convention these days let me know, but it's been a good convention for differentiating them from decimal numbers)

My ideas of using base6 for everything all started with Naccel , but expanded into more than a pipcount at some point.

The first trigger was that 6's translate perfectly to dice rolls. Since we use 6 faced dice in bg every step in future estimations translates into multiples or powers of 6. Those numbers we always end up using 36, 216, 1296, or even the bigger ones that we probably give up before using all translate an even 1 when youfre in base6. What that means is that any denominator that has a value which is a power of 6 (for example, probably all dice calculations) can just be thrown out and there is no denominator at all and the bigger it gets, it only adds some less insignificant digits tacked onto the end and still no denominator.

I'll show you what I mean.

9 times out of 36 is a somewhat common theme. If something is to occur 9 times out of one dice roll, It turns out it's exactly 25% well, if you convert 9 to our base6 notation it can be expressed as 1:3 so now we have a reference, 1:3 chances means 25% for a single dice roll but this is where it gets interesting, if we have a case that happens 54 out of 216 we can do the division and find out that it is also 25% , but if we express the number 54 in base6 we get 1:3:0 , 324 out of 1296 is also a 25% number , 324 is 1:3:0:0

if you haven't recognized what's happening
9/36 -> 1:3
54/216 -> 1:3:0
324/1296 -> 1:3:0:0
1944/7776 -> 1:3:0:0:0

and on and on right up to numbers that you really wouldn't want to think about in decimal. This consistency means we donft ever have to deal with fractions, and that means we donft have to do any division after all that multiplying.

So no matter how many levels deep you want to go, if you do your multiplication in base6 you will never need a denominator ie. you can just multiply the number of shots etc. that you are counting up and thatfs your number. There is no need to then divide bigger numbers into it etc to find your reference, youfve already got it. One key trick is to multiply your numbers from the head down (not from the tail end like you learned in school) so you get the significant digits first then stop once you get to 2 or 3 digits and you may have saved a LOT of work. There are math tricks for muliplying that make things much quicker but the first point is to get used to multiplying head down rather than from the tail.

The next step in this pursuit is understanding how percents can be converted. You can think of it as a continuum 216 long rather than 100 (then also expressed in base6) so it follows that when you convert the number a 1% change will move it 2.16. The long way to convert a decimal number is just by multiplying by 2.16 then converting it to base 6, but if you donft like doing the multiplication every time, you may be able to do better by having some reference points then adjusting by 2.16 a pop for numbers that are off the mark from your references.

Here are some useful reference patterns
pattern (100 / (4/x))
1:3:0 is 25%
3:0:0 is 50%
4:3:0 is 75%

pattern (100 / (5/x))
1:1:1 is 20%
2:2:2 is 40%
3:3:3 is 60%
4:4:4 is 80%
5:5:5 repeating is 1:0:0 = 100%

pattern (100 / (6/x))
1:0:0 is 17%
2:0:0 is 33%
3:0:0 is 50%
4:0:0 is 67%
5:0:0 is 70%
6:0:0 is 1:0:0 = 100%

So if you are trying to convert a percent to this convention you can find the closest match then adjust by 2.16 for each 1% it is off. For example if your number is 22% you will already know 1:1:1 is 20% then add 4 plus change to it for 1:1:5 I had previously come up with a couple of formulas for approximation but they tend to take more effort than this interpolation and you donft really gain any accuracy.

Next, I converted my match equity table all into base6 You can memorize the table but basically how I use it is I just have a Neilfs Numbers conversion that uses these base 6 numbers rather than decimal numbers. This way I can get the match equity numbers for any score and figure out takepoints , gammonprice etc. and have them all come out in this form so that shot counts , winning chances etc. are all integrated and there are no lumbering denominators. Itfs pretty neat package.

For a simple example to see how you might use these numbers:

Say you are 4 away 3 away and he throws the 4 cube at you.

You would figure out what you need the same way as you already do. It could go something like: This 4 cube goes off both ends for the win and loss (0% and 100%) so simply taking the equity at the score after a pass (4 away 1 away) should give you your takepoint. That number is about 1:0:3 on the MET (around 18% in decimal) So letfs see if you have enough to take it.

Your opponent has 2:5 misses (17) and 3:1 hits (19) next roll (He misses on all 1fs, 23, 24, and 34) Then there is 21 twice which brings his misses up to 2:5:0:4 , the extras from the 21-21 case could be easily used even mentally over the board but the intent of this post is just to walk through the ideas so wefll use 2:5 for the miss number and keep it to 2 digits.

You have 3:4 misses (22) (any 1 or any 2 and 34 (there are a few residuals which could easily be added just by making sure they are on the correct digit from the top, but wefll ignore them as above so we can follow a simple 2 digit number for the sake of the example)) you can invert that 3:4 to hits by subtracting it from 1:0:0 (36) and you get 2:2 (14)

So if we want to find our approximate chances to win we multiply his 2:5 by our 2:2 The basic technique for multiplying may look something like below when we have to show it on paper, but in practice you can do just do it from the top and add and carry the digits as needed.

Multiply the first digits 2 * 2 = 4
Cross multiply the middle digits and add them 2 * 5 = 1:4 2 * 2 = 4 1:4 + 4 = 2:2
Then multiply the last digits 2 * 5 = 1:4

When you compile them:
4
2:2
0:1:4
Added together gives you 6:3:4 or properly expressed as 1:0:3:4

So you need 1:0:3 to take and youfve got 1:0:3:4 , thatfs pretty close Maybe that 21 twice isnft so trivial after all (good thing it plays in the favor of the take otherwise wefd regret tossing it out and may want go back with another calculation to see if it doesnft cross the line) If the score happened to be reversed, 3 away 4 away, it still goes off both ends so the equity of the score after a pass (3 away 2 away) would be 2:2:2 rather than 1:0:3 and it would be a huge pass if we were only getting 1:0:3:4

There are many situations we can go into where it becomes very useful to use these numbers, but, there are also some significant drawbacks to doing this. Some drawbacks of using this include

1) The obvious , you have to practice counting and multiplying in base6
2) No matter how good you get at these calculations the question that every player will ask is still not resolved. That is g So how do you know just how much equity you have in a position ? g I guess what Ifm getting at is that you can spend an awful lot of time getting good at this and it may not improve your backgammon nearly as much as the same time investment in actually learning positions.
3) This is probably the big one. Every backgammon book , backgammon software, and most other players discuss positions in terms of percentages (or decimal based equity) so you still have to convert everything in order to communicate

So in the end, the cost of taking these senary ideas all the way to match equity may or may not be worth the effort but there are certain times when it is very powerful.

Naccel, on the other hand, has great value because of itfs lightning speed and ease of effort so itfs well worth pursuing in itself. For me , itfs one of the most outstanding tools in backgammon.

Don T

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