I. Introductory Notes to Lotto Wheels: The Mathematical Foundation of Lotto Wheeling
Let's take the two most basic sets and wheel their elements: 9 numbers, and 12 numbers, respectively. I sold (very cheaply!) this wheeling method and also a good set of lotto wheels in 1986-1987. I discontinued the offer because I was able to buy an IBM-compatible PC and started to write serious lottery software, including for lotto wheeling. I did a Samaritan thing. I had a number of copies of my cheap lotto-wheel brochure left. I gave it away at a gas station in Gettysburg, Pennsylvania. The station was also a licensed lottery agent (I played there sometimes, when I fueled my car). I returned to the same station in a couple of weeks. The lottery agent asked me if I had more copies of my lotto wheels brochure. She said: "It went like hot cakes! Got more copies?" NOT! Unfortunately, I didn't even save one copy for myself!!! The lotto wheel leaflet was typed. I didn't have a personal computer at that time (only very few individuals had; by far, most of them were living in the United States).
You might be able to find someone who bought my 16-page lotto wheel brochure. I charged between $3 and $5 for the useful package. I was told (hush-hush!) that my material is a collectible now. If you are really interested, just google for a while Also, you might have heard of one Robert Serotic. He too was an immigrant to the United States. He published books on the topic of lotto wheels. Robert Serotic published also a lotto wheeling newsletter. A user of my lottery software surprised me when he sent me Serotic's newsletter. Robert Serotic had published without permission the lotto wheeling method presented in my brochure. Granted, he gave me credit as the author. Problem was, he printed my method secretly, without asking for my permission (forget about pay!) Better Business Bureau of California took notice of the incident!
1) 9-number wheel for 6-number lotto games.
We can divide the numbers in 3 groups of 3 numbers each: A, B, C. Group A=1,2,3; group B=4,5,6; group C=7,8,9. We can combine the 3 groups 2 at a time in a total of C (3, 2) = 3 combinations of 6 numbers each: AB, AC, BC. Converting the groups to their initial compositions, we get:
1,2,3, 4,5,6
1,2,3, 7,8,9
4,5,6, 7,8,9
The lottery commission draws 6 winning numbers. The drawing is random, therefore the 6 winning numbers will have a random spread among the 3 3-number groups. There are 3 distribution possibilities:
0-3-3 = 6 winners of 6 = 33%;
1-2-3 = 5 winners of 6 = 33%;
2-2-2 = 4 winners of 6 = 33%.
Since all the groups are paired with one another, our 3-combination lotto wheel cannot have fewer than 4 winning numbers. They say, the lotto wheel assures a so-called 'minimum guarantee': '4 of 6'. That is, if the 6 winning numbers are among the 9 we selected, the wheel guarantees that at least one combination will contain at least 4 of the winning numbers. In 33% of the winning distribution possibilities, the 9-number lotto wheel hits the jackpot ('6 winners of 6')! Moreover, this lotto-6 wheel for 9 numbers also guarantees '4 of 5'. There are no worse distributions of the 5 lotto winning numbers than 1-2-2 or 1-1-3.
2) 12-number wheel for 6-number lotto games.
We can divide the numbers in 4 groups of 3 numbers each: A, B, C, D. Group A=1,2,3; group B=4,5,6; group C=7,8,9; group D=10,11,12. We can combine the 4 groups 2 at a time in a total of C (4, 2) = 6 combinations of 6 numbers: AB, AC, AD, BC, BD, CD. Converting the groups to their initial compositions, we get:
1,2,3, 4,5,6
1,2,3, 7,8,9
1,2,3, 10,11,12
4,5,6, 7,8,9
4,5,6, 10,11,12
7,8,9, 10,11,12
The lottery commission draws 6 winning numbers. The drawing is random, therefore the 6 winning numbers will have a random spread among the 4 3-number groups. There are 5 distribution possibilities:
0-0-3-3 = 6 winners of 6 = 20%;
0-1-2-3 = 5 winners of 6 = 20%;
0-2-2-2 = 4 winners of 6 = 20%;
1-1-2-2 = 4 winners of 6 = 20%;
1-1-1-3 = 4 winners of 6 = 20%.
Since all the groups are paired with each other, our 12-combination lotto wheel can't have fewer than 4 winning numbers. Thusly, the lotto wheel assures a so-called 'minimum guarantee': '4 winners out of 6'. That is, if the 6 winning numbers are among the 12 we selected, the wheel guarantees that at least one combination will contain at least 4 of the winning numbers. In 20% of the winning distribution possibilities, the 12-number lotto wheel hits the jackpot ('6 winners of 6')!
Those two reduced lottery systems are the only lotto wheels totally based on solid mathematics. The two systems offer tremendous leverage. Personally, I will never play any other lotto wheel. Actually, I settled on the 12-number wheel only. The 12-6 lotto wheel provides very efficient leverage. This extraordinary phenomenon in lotto wheeling is known as the Parpaluck effect. It only occurs in lotto games that draw an even amount of winning numbers: Lotto-4, lotto-6, lotto-8 (if ever implemented, etc.) For example, lotto-4 games draw 4 winning numbers. The optimal wheel has 8 numbers (the double of the winning numbers). Divide the 8 numbers in groups of 2; a total of 4 groups results; wheel the 4 groups two at a time; the final result is a 6-line lotto wheel, with 4 numbers per combo. The distribution of the winning numbers in the 4 groups can be: 0-0-2-2 (first prize); 1-1-1-1 (the worst distribution = third prize); etc.
These are the optimal cases of wheeling lotto numbers. No, the lotto wheels cannot be created equal! Take the 12-number situation. We can group the numbers 2 at a time, for a total of 6 groups of 2 numbers each. We combine the 6 groups 3 at a time, for a total of C(6, 3) = 20 combinations, 6 lotto numbers at a time. The distribution of the 6 winners gets worse, like in this case:
1-1-1-1-1-1 = 3 winners of 6!
Now, we constructed a larger lotto wheel 20 6-number combinations but it only guarantees '3 winners out of 6'! Pay more to get less! This negative phenomenon in lotto wheeling is known as the Kokostirk effect.
We all know of another popular lotto format: Drawing 5 winning numbers. How about them 5-number lotto wheels? Short answer: No, we can't construct the same high level of efficiency as in the case of 12 number lotto wheels drawing 6 winning numbers. But we can come acceptably close. Again, we must put the numbers together, in groups. I built a 5-number wheel on the same mathematical principles.
The double of the 5 winning lotto numbers is 10. Therefore, the optimal number pool must have 10 elements. Drawback: We can't arrange the numbers in equal-size groups (because 5 is a prime number). So, my best method was to create two groups of 3 numbers each, and two groups of 2 numbers each. Group A=1,2,3; group B=4,5; group C=6,7,8; group D=9,10. We can combine the 4 groups 2 at a time in a total of C (4, 2) = 6 combinations of 5 numbers: AB, AC, AD, BC, BD, CD. We deleted one number from the 6-number resulting combination; also, we added one number to the resulting 4-number combonation (a favorite term of mine). Converting the groups to their initial compositions, we get:
1,2,3, 4,5
1,2,3, 6,7
1,2,3, 9,10
4,5, 6,7,8
4,5,8, 9,10
6,7,8, 9,10
The random distribution of the 5 winning numbers in a lotto-5 draw is more complicated than in a lotto-6 game. We can't group the numbers equally. Thus, the 3-number group can have 3, or 2, or 1 of the winners in a drawing.
0-0-2-3 = 5 winners of 5 = 14+%;
0-1-2-2 = 4 winners of 5 = 14+%;
0-1-2-2 = 4 winners of 5 = 14+%;
0-1-2-2 = 4 winners of 5 = 14+%;
1-1-1-2 = 3 winners of 5 = 14+%;
1-1-1-2 = 3 winners of 5 = 14+%;
1-1-1-2 = 3 winners of 5 = 14+%;
I believe this 5-number lotto wheel has good leverage too. For starters, it assures the '3 of 5' minimum guarantee alright. In fact, the '3 of 5' prize is offered in 43% of the favorable cases (when our 10 picks contain the 5 lotto winners in a drawing). In 43% of the favorable cases, this lotto wheel assures at least one '4 of 5' winners. Not to be neglected, this lotto wheel assures '5 of 5' winners with a 14% degree of certainty.
II. Tools of Lotto Wheeling: Mathematics, Theory, Software
The above is a new sub-set, constructed from the inclusive set C(12,6)=924 combinations of 6 numbers each. In my theory of sets, if the inclusive set has a main completion rule, a sub-set can be constructed if and only if it has a specific completion rule derived from the main completion rule.
Knowledgeable players will not play combinations such as 1,2,3,4,5,6. I detailed in another message how wasteful such play really is. The players will usually select their own numbers, also known as 'picks'. For example: 7,13,24,8,33,35,42,28,19,37,40,49. The next important step is wheeling the picks. That is, the player needs to replace the theoretical numbers 1,2,3,4, 12 by the picks 7,13,24,8, 49. The process can be done manually, but it is tedious and error prone. The players use software to wheel their lotto numbers. Software wheeling cost money, up until I released freeware to wheel lotto numbers.
Previous software wheelers were also limited in scope. They could only wheel their own abbreviated lotto systems. Generally, they were unable to wheel third-party wheels or Powerball systems. My lottery wheeler software is the most comprehensive to date. It can wheel any kind of wheel, for up to 100 numbers per combination. It can also wheel Powerball, Mega Millions and Keno lotto wheels or systems.
Enter the scene:
FillWheel.EXE - August 2002 - Freeware.
LottoWheeler.EXE - September 2006 - The best - Free 32-bit lotto wheeling software.
That's the easiest way to convert the theoretical lotto wheels or systems ('1,2,3,4,5,6' etc.) to combinations of real picks ('29,13,33,2,44,39' etc.)
A few more wheels from yours truly. They were created by my lotto software WheelCheck6.exe (also available for 5-number lottos: WheelCheck5.exe):
* 10 numbers, 100% '4 of 6', 3 combinations (exactly the lotto odds)
1 2 3 4 5 6
1 3 4 8 9 10
2 5 6 7 9 10
* 11 numbers, 100% '4 of 6', 5 combinations (20% over the lotto odds)
1 2 3 4 5 6
1 2 8 9 10 11
1 3 5 8 10 11
2 3 5 7 9 10
4 6 7 8 9 11
Nobody had been able to construct such lotto wheels before my software. The two wheels, however, mushroomed all over the Internet, a few months after I published them!
Be aware, however, that the lotto wheels can do more harm than good. Read my article "The myth of lotto wheels or abbreviated lotto systems" (follow the link section). The player must use a good strategy of picking lotto numbers, before using lotto wheels. Please read the main lotto lottery strategy page at this site. My own software, LotWon, SuperPower and MDIEditor and Lotto WE comes with a number of lotto wheels or systems. Please keep in mind that those systems were not optimized according to my own set theory. The wheels contain more combinations than the respective lotto odds. Such systems aimed at hitting better prizes. They are meant to be used in conjunction with the lottery strategies I have presented at this website.
The 6-40 lotto game back then (1986) required to play at least two tickets for $1. We played a total of 36 combinations, for $6 each. I had won twice before with one of my lotto colleagues ('4 of 6'). That time, I applied my 9-number lotto system. The three of us in the lotto group also won once another '4 of 6' third prize. That prize usually paid over $100 per winning ticket.
We played 12-number lotto wheels. I used two wheels for the same 12-number group. The first lotto wheel used the group in lexicographic order, from 1 to 12, in 6 combinations. Then, I shuffled the 12-number set and applied the wheel to it. I always used the same shuffled set to save time. In total, we played 3 groups of 12 numbers each.
The program I wrote generated 12-number combinations randomly. The numbers were not sorted: RAM, speed, and retyping were the main reasons. I wrote the last 20 lotto drawings at the beginning of the program, in the READ/DATA section. I hadn't seen any 4-number group repeating from the last 20 drawings. Of course, the average no-repeat is much longer than 20 draws. Trying to eliminate all 4-number groups in 12-number combinations would be a daunting task (virtually, mission impossible, given the home-computing technology at that time). So, I had to be selective. I eliminated 1-2-3-4 and 9-10-11-12 for sure; then, a few more groups in between, like 4-5-6-7, 8-9-10-11.
The computer (Atari 800XL) had a real hard time! I let it run until I saw at least 10 combinations on screen (no disk to save the output!) I wrote down the last three 12-number combos (2 lines each on screen). Then, wheel each 12-number lotto combination twice (manually). It was laborious (you can try it for yourself) and it took some time. Then, fill out the play slips. There were some errors every time, especially filling out the lotto play-cards.
Read Ion Saliu's first book in print: Probability Theory, Live!
~ Founded on valuable mathematical discoveries with a wide range of applications, including mathematics of lotto wheels, and manual creation of lottery wheels.
III. Resources in Lotto Software, Lottery Software, Lotto Wheeling.
It lists the main pages on the subject of lottery, lotto, software, wheels and systems (click above).