## Congruence Usage

pick4victory
Getting used to it Posts: 5
Joined: Tue Nov 04, 2014 10:53 am

### Congruence Usage

Hi,
I am very new to LA and very confused. I was looking at Congruence 9 and it shows like below for the Pick 3 Game:

PICK3 NUM [C] - Congruence 

4-4-2 8/1/1/0
5-5-0 8/1/1/0
5-2-4 7/3/0/0
5-6-2 7/3/0/0
6-9-1 8/1/1/0

ALL over the history file, only these 2 congruence can be found. Does that mean I can assume for the next draw, if I target 7/3/0/0 patern is going to come, how do I compute the next numbers ? is the correct way to look at it ? appreciate your insights....I am confused and could not understand the manual.

lottoarchitect
Posts: 1555
Joined: Tue Jan 15, 2008 5:03 pm
Location: Greece
Contact:

### Re: Congruence Usage

Hi pick4victory, I can't recall exactly how the congruence is computed for Pick style games (it has been several years) but when I'll have the time I'll get back to you on how exactly these are computed. However, what we care about any rejection filter, including the congruence ones, is to identify some pattern and possibly estimate the outcome of the next draw regarding that filter. To have an idea of the possible values a filter can return, just open the rejection filters design window (F8) and check the values indicated as YES/NO. Any value that has a YES/NO, has at least one possible ticket that returns that value. You simply then set the values you believe they'll not come at the next draw to NO and save that filter as per the instructions at the other post. So, even if you don't know really what a rejection filter computes, the usage logic is universal.
If you believe the 7/3/0/0 will come out (this is 4 different filters!), by all means set up it that way at the rejection filters and proceed to stage 3 calculations to produce the tickets.
I'll get back to you when I examine in detail exactly how congruence is computed in Pick style games, if you are interested in that detail.

pick4victory
Getting used to it Posts: 5
Joined: Tue Nov 04, 2014 10:53 am

### Re: Congruence Usage

Thanks for prompt reply on both of my posts. I will try the steps. And yes, by all means, if you could find out for me how the congruence works for pick3, I would really appreciate that as I am very curious to see the pattern repeat draw after draw ..thanks.

lottoarchitect
Posts: 1555
Joined: Tue Jan 15, 2008 5:03 pm
Location: Greece
Contact:

### Re: Congruence Usage

Ok Pick4Lottery, I have examined the Congruence filter, it is easy actually once you get used to it.

First there are 3 types of congruence filters, the normal, the [ S ] (shape) and [C] (counter) types.

I'll explain first the normal type, since the others are derived from it.
The idea behind congruence is to convert a number to one digit only, based on the modulo X it returns (the remainder).
For lotto style games (cannot draw same numbers), the conversion can produce a value 1-X (as per the example at the help file, any modulo which results to 0 is assigned to X). For Pick style games (same numbers can be drawn), due to the number 0 that can be drawn, the range of possible values is 0-X.
Therefore, for Pick style games, you'll always get one more column displayed to cover the 0 result.

Let's examine the outcome of Congruence  using a draw, where the X = 3 (the modulo). The exact same process is used in pick-3 like your game too.

For Pick style Congruence[X], the possible values are always 1 to X for all numbers, except 0 which results only with number 0 (so for X=3, the range is 0-3, 4 possible values)
A Pick-4 draw is 3 4 0 4 as Congruence
Number 3 mod X (3) = 3 (3 mod 3 = 0 and 3 is different to 0 therefore the result is X)
Number 4 mod X (3) = 1 (4 mod 3 = 1)
Number 0 mod X (3) = 0 (0 mod 3 = 0)
Number 4 mod X (3) = 1 (4 mod 3 = 1)

If you display this as rundown statistics, you'll see 1 / 2 / 0 / 1 because there is 1 0-value, 2 1-value, 0 2-value and 1 3-value.

So the 0-value of any Congruence[X] is produced only if the number tested is 0. Any other number that results in n mod X = 0, is assigned to the value X.

For lotto style Congruence[X], we don't have the value 0 as a possible outcome
A lotto draw e.g. 3 15 19 40 45 as Congruence 
Number 3 mod X (3) = 3 (3 mod 3 = 0 -> result = X)
Number 15 mod X (3) = 3 (15 mod 3 = 0 -> result = X)
Number 19 mod X (3) = 1 (19 mod 3 = 1)
Number 40 mod X (3) = 1 (40 mod 3 = 1)
Number 45 mod X (3) = 3 (45 mod 3 = 0 -> result = X)

If you display this as run down statistics, you'll see 2 / 0 / 3 because there is 2 1-values, 0 2-values and 3 3-values.

In Lotto style games we don't have the number 0, therefore the 0-value will never show up and this is why we don't have the 0-value column at the congruence display.

In exactly the same manner works any other Congruence[X], where X denotes the mod X used.

The [ S ]-Congruence is explained at the help file and each column of that filter corresponds to the mod X value used (X has the range 3-9 always) therefore you'll always get 7 columns to that filter.

The [C]-Congruence[X] filters belong to the general concept of "Counter" filters (this is what the [C] part denotes) and any [C] type filter requires the existance of the relevant normal filter, in this case, in order to compute the [C]-Congruence, we must have Congruence too.
The example you ask above is a Counter filter and I'll display the process on how these results have been produced. I'll do it only for the first draw 4-4-2 8/1/1/0

First, let's analyse 4-4-2 as Congruence
Number 4 mod X (9) = 4
Number 4 mod X (9) = 4
Number 2 mod X (9) = 2

So, if you display this as rundown statistics you'll get 0 / 0 / 1 / 0 / 2 / 0 / 0 / 0 / 0 / 0 because we have 1 2-value and 2 4-value.
So, what we have here is 0 / 0 / 1 / 0 / 2 / 0 / 0 / 0 / 0 / 0, which has 8 0's, 1 1's and 1 2's.
If you notice, this is what the [C]-Congruence displays as 8/1/1/0. The last 0 comes from the fact that is is also possible to have a 3 as possible outcome e.g. the pick 3 draw 7 7 7 as Congruence results in 0 / 0 / 0 / 0 / 0 / 0 / 0 / 3 / 0 / 0 and we have 9 0's and 1 3's so this will produce [C]-Congruence as 9 / 0 / 0 / 1.

What I demonstrated above is what the help file also says in one sentence (extracted from the help file):
Thus, the counter filter counts how many columns from the related basic filter return a given value (which is a column in the [C]-x filter).
So, seeing this in a different view, [C]-Congruence resulting to 8/1/1/0 means:
8 columns of normal Congruence resulted in value 0
1 column of normal Congruence resulted in value 1
1 column of normal Congruence resulted in value 2
0 columns of normal Congruence resulted in value 3

One side-note: If you sum the result of a [C] type filter, you must always get the total columns displayed at the relevant normal filter. Since Congruence produces 10 columns (0-9), sum of 8 + 1 + 1 + 0 = 10.

[C] type filters are very powerful because you don't need to specify which columns of the normal filter must return a specific value, only how many of them will return a value.

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