# Positional Lotto Wheels, Reduced Lottery Systems Position-by-Position

## By Ion Saliu, Founder of Lotto Wheeling Mathematics

Published in January 2018.

Axiomatic one, you might be among the visitors who asked me about positional lotto wheels, especially starting the year of grace 2017. The interest grew by an order of magnitude after I posted analyses of positional lottery wheels in the venerable rec.gambling.lottery (RGL) newsgroup. My in-position lotto wheels showed a tremendous advantage over any similar systems. My thread in the Google groups is still live:

The 274-line position-by-position lotto wheel hit one 4 of 6 winner and 8 3 of 6 winners. It happened on January 24, 2018.

My lotto software can also check for winners in positional groups of numbers without expanding them to combinations first. The positional frequency system hit two jackpots in one year worth of lottery drawings. The un-wheeled system recorded also 15 match-5 and 23 4 of 6 winners. I didn't check for match-3  there would have been plenty of those! The positional systems are the way to go in lotto. They hit more often (more efficiently) because the lotto numbers have an inherent tendency to hit in certain positions. The positional lotto systems (based on both frequency and skips) respect that natural tendency.

I haven't written, however, dedicated software to generate lotto wheels from ranges of numbers. That is, given, say, 6 strings of numbers for the lotto-6 games, let's generate lotto wheels where the numbers are strictly in-position. If a lotto number is only in the string in position 1, that entity must always appear only in the 1st position of the combinations. No 2 or more numbers in the same position will appear in the same combination. In the real system you see below, the lotto numbers 1 and 4 in the first-position string will not show together (in the same combination).

Fear not! My software is more than capable of handling a task of such magnitude  and it is the only software of this kind. It only takes two steps to generate positional lottery wheels from output files. Until I decide to write specialized, in-one-step positional-wheeling software, you can simply follow the easy procedure clearly described on this Web page. Not to mention that the personal computers are very fast nowadays and they can fulfill this task with flying colors.

### STEP 1: Generate combinations from strings of numbers arranged by positions

The 6-number lotto game has 6 positions; therefore you'll have 6 strings of numbers. Example: the positional frequency system we have analyzed in RGL. The system consists of the top-12 UK 6/59 lotto numbers in each position (based on frequency).
```  2   1  10  11   4   8   5   7  13   3   9  12
14  17  12   3  13  11   8  22   7  15  16   2
30  31  26  18  25  16  20  29  21  22  35  27
32  34  37  46  38  39  42  28  40  41  19  24
40  51  44  41  53  48  49  43  52  47  54  37
58  57  59  54  52  55  56  50  51  49  42  46
```

In this case, each string has an equal length (same amount of numbers). But the strings can also be uneven, as it is mostly the case with the positional skips systems.

Software to generate lotto combinations strictly position-by-position: SkipSystem, menu II of the Bright / Ultimate grandiose lottery applications. Functions: S = Lotto 6, then G = Generate Combinations from Systems.

• Additional operation (not mandatory, but strongly recommended): Apply the LIE Elimination in a combination generator in my software. For example, you might want to eliminate combinations generated by non-positional systems as they hit far more rarely. You might want to take a look at my first post in the aforementioned RGL thread.
• First, I generated combinations from the non-positional system consisting of the top-24 UK 6/59 lotto numbers (based on frequency). As you know by now, that is, after reading my theory, the non-positional lottery systems fare significantly worse than their positional counterparts. Therefore, they represent ideal candidates for LIE Elimination.
• Next, we generate combinations from the positional systems. They hit far more frequently, albeit with many more combinations. But that's why we employ the LIE Elimination strategy!
• Finally, we LIE Eliminate the combinations generated by the non-positional string of numbers from the combinations generated by the positional frequency lottery system. In my example, I applied the LieID filter = 4. Translation: My level of confidence is high that the non-positional system will not fare better than 3 of 6. That is, more often than not, the non-positional number will not hit 4, or 5, or 6 winners. In the case above, the situation was way, way better. The non-positional frequency system has one and only one hit. Thus, I could have applied the LieID filter = 2 and very few combinations are generated. The next step  wheeling  would not be necessary. We would hit as many winners as the positional system hit unexpanded.
• There are many more reduction strategies one can apply with high degrees of certainty. Both frequency and skip-based non-positional systems can be fed to LIE Elimination, one file at a time. The LieID filters might be different, however.
• If the LieID filters are equal for various files, you can concatenate (combine them in one file) and perform the reduction in one step.
• The LIE Elimination function is present in all combination generators of my lottery software.
• You can learn a lot more by reading the specialized page in the Resources section. In fact, every entry in Resources is really valuable.

### STEP 2: Wheel the output file generated at Step 1

• Additional operation (not mandatory, but strongly recommended): Shuffle or randomize the final output after the LIE Elimination.
• Pertinent software: Shuffle.exe, menu IV of Bright / Ultimate software, function R = Randomize, Shuffle Elements. Then, select the function F = Shuffle Files (Numbers & Text) in the program.
• Wheeling a randomized file does bring about better results: Fewer lines in the wheel while maintaining the minimum guarantee.

You have three choices here. The first one was applied in the aforementioned thread.

A) I pointed you to one of the functions (programs) in my lottery software capable of such much-sought-after task: Super Utilities (SoftwareLotto6.exe). It is listed as Super Utilities on the main menu of the Bright / Ultimate grand lottery packages. We'll use for this task the function D = Duplicates: Strip and Wheel. I chose the 3 of 6 minimum guarantee to get the minimal amount of combinations to be published in the Google group.

B) Function H = Wheels from Files (WheelIn6.exe), menu #2 of the Bright / Ultimate great applications. If I choose the 3 of 6 minimum guarantee again, I'll get the same amount of lotto combinations as in Option A.

H = Wheels from Files is very similar to D = Duplicates: Strip and Wheel.

C) The best wheeling choice: W = Wheel On-the-Fly (Wheel6.exe), also on menu #2. In Wheel6, select the minimal guarantee, and then the function P = Purge-Wheel an Output File. This time, however, if I apply the 3 of 6 minimum guarantee, I'll get fewer combinations than in Option A or Option B. Therefore, the lottery wheel is more efficient cost-wise.

This is the most efficient option. It generates random combinations that never start at 1, 2, 3, 4, 5, 6 or 44, 45, 46, 47, 48, 49. Also importantly, the amount of combinations generated is lower than in A or B  without affecting the minimum guarantee. There is a higher chance at hitting higher prizes. You can even apply safe filters to reduce the total of lines generated and still preserving the minimal guarantee.

There is one drawback, however: Speed. You need be more patient compared to options A and B. If you wait more than half an hour without a single combination having been generated, you can stop the program. It probably found a complete randomized lotto wheel that satisfies the minimum guarantee.

### World-Record Lotto Wheel (54, 3 of 6) = 274 Lines

The special reduced lottery system was created in the process unintentionally. The previous world record in lottery wheeling was for 49 unique numbers in 163 lines. Not to mention that the particular cover or design had a severe flaw: It was a SPLIT wheel. The lotto numbers were divided into two groups; the numbers in one group were never combined with numbers from the other group. No wonder such lotto wheel fares very poorly when targeting higher-tier prizes.

My cover or lotto design treats the numbers equally. The key feature is the natural bias of the lotto numbers toward one or two positions in the combination. Evidently, nobody has ever seen a 6/59 combination to start with number 58; and we'll never see lotto number 3 in the 6th position!

How can I state I created the world-record lotto wheel? I can answer such a question and set it on a solid foundation. I can compare the sizes of the two designs and how many combinations they cover.

The previous world-record lotto wheel: 163 lines; my design: 274 lines. Ratio: 274 / 163 = 1.68. The previous design covered 6 of 49 numbers or 13983816 combinations. My lotto wheel deals with 6 of 54 unique numbers that generate 25827165 combinations. Ratio: 25827165 / 13983816 = 1.85.

Advantage of my lotto wheel: 1.85  1.68 = 0.17 (17%). Therefore, my lotto design is the champion of the world by a margin of 17 percentage points. For parity, the (49, 3 of 6) lotto wheel should have a maximum of 136 lines, not 163.

[Lotto] wheel in the sky keeps on turnin'
I don't know where I'll be tomorrow
Wheel in the sky keeps on turnin'.

Journey, Wheel In The Sky

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