COMBINATORIAL SYSTEMS (WHEELS) WITH GUARANTEED WINS FOR PICK-5 LOTTERIES
INCLUDING
EURO MILLIONS AND THE MEGA LOTTERIES

by Iliya Bluskov, Ph.D.


NEW EDITION (2020)


        Copyright 2020 Iliya Bluskov, Lotbook Publishing ©            

 

New Pick-5 book - cover


All rights reserved under the Pan-American and International Copyright Conventions. This material may not be reproduced, in whole or in part, in any form or by any means electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system now known or hereafter invented, without written permission from the publisher, Lotbook Publishing.

 

All inquires should be addressed to:

Lotbook Publishing
P.O. Box 33031
West Vancouver, B.C.
V7V 4W7 Canada

       or e-mail: lotbook@telus.net

In his new pick-5 book, Dr. Bluskov’s once again delivers the best known strategies to the lottery players; this time his focus is on Pick-5 lottery systems. Players who are interested in systems for Pick-6 lotteries should check his pick-6 book. In this presentation of the pick-5 book, some sections are omitted, others are partially presented. The Contents lists the titles of all sections except the titles of the systems; three (out of 126) systems are presented here. The full text and the remaining 123 systems are in the book.

 

A note from the publisher

 

Contents

 
Foreword
Introduction

 Lotteries

 What is a lottery system?

 Guarantees

 Minimality

 Tables of wins

 How to use the systems

 Choosing your numbers

 Can you win the top prize in your lottery by playing with a system?

 The odds of winning

 Bonus number(s)

 Playing systems in the big double-pick lotteries: Wheeling the bonus number(s)        

 Wheeling bonus numbers in Euro Millions

 Finding your way through the book

 Navigation table by number of combinations

 Navigation table by quantity of numbers

  PART I: Lottery systems with a single guarantee

 System # 26: 9  numbers, 12 combinations

 (and 105 other systems)

  PART II: Lottery systems with double guarantees

 System # 108: 9 numbers, 18 combinations

 (and 10 other systems)

  PART III: Lottery systems with multiple guarantees

 System # 126: 11 numbers, 11 combinations

 (and 8 other systems)

  PART IV: More about the book and systems

 Is this the best book on the market?

 How a system is constructed

 Larger systems

 Acknowledgments

 Comments, reviews, testimonials
Ordering Information

 

 

 

A note from the publisher

 

        Dr. Bluskov is a university professor. He has also worked as a freelance writer for a number of lottery publications in Europe and North America, and has spent many years in research in Combinatorics, a branch of Mathematics, which, among other things, deals with combinatorial lottery systems. He turned his lifelong fascinations with numbers and structures into ground-breaking strategy books for lottery players. Part of his research is on determining how to achieve a specific winning guarantee in a lottery in the minimum possible number of tickets. The results in his books originate from recent research in several areas of mathematics, and have been obtained by various techniques ranging from sophisticated combinatorial constructions to applying optimization methods via thousands of hours of computations. Although based on quite advanced knowledge in mathematics and related areas, the results are described in plain language and are accessible to virtually every lotto player or a group of players. Dr. Bluskov’s lotto books have been the best on the market for a long time, and we firmly believe he will keep it that way for the years to come. We hope you will agree with that statement when you go through the content and use some of the systems. Dr. Bluskov is the only author who has actually created world record breaking lotto systems. His books present a number of improvements and new features that cannot be found in any other book on lotto systems. These include the organization and the presentation of the material, the systems with multiple guarantees (never appeared before in a lottery publication) and the complete tables of possible wins. Each table has been generated by a complete experiment over ALL theoretically possible draws. A computer program has performed the experiment for every system in the book. The tables have been additionally abbreviated and custom designed "by hand". The complete tables appear also for the first time in a lottery publication. Wheeling the bonus number(s) in the Mega lotteries (Lucky Star numbers in EuroMillions) is completely new to the genre. Dr. Bluskov was the first to introduce and study several ideas that lead to establishing a number of properties defining a good lotto system. These include minimality, balance, and maximum coverage, features that are discussed in the text. Information about the availability of these features is provided in his comments for each system in his books. The book presented here is for pick-5 lotteries and contains 126 systems (wheels) all of which are the current world records in terms of providing any particular guarantee in the minimum known number of combinations.

 

 

Foreword

  

 

Readers are often interested to know whether I play the lotteries, how I play and how much I have won. Indeed, I do play, and when I play I do use systems. I used to play a lot when I was younger. Nowadays, I have a good day job that keeps me busy even at nights, I am involved in publishing books, both lottery and mathematical ones and invest inProf. Bluskov the stock market; overall, I do not really need to win a jackpot to have a comfortable life. Still, I do not mind winning one! I do not think there is a single person on earth who minds winning one either. I have always looked at the lottery as an entertainment, not as a source of income. To get rich from the lottery you have to be among the few lucky big winners; in other words, you must  be a jackpot winner, either alone or with a group of people. The purpose of my book is to show you, the readers, the strategies that many big winners, syndicates and individual players alike, use regularly to play the lottery, scooping some small wins along the way, while waiting for the big hit. You might be one of them one day, it can happen.

I would like to make it perfectly clear that I do not promote any particular lottery and am not connected to any of the existing lotteries. The book itself does not promote playing with a (large) number of tickets. It is a book for those players (or groups of players) who have ALREADY decided to play with more than two tickets. You are probably interested in this book, because you are going to throw a couple of bucks at the next jackpot anyway, right? So, my question is: Why not try the strategies of the big winners? This book will lead you through the widely used highly entertaining and precise strategy of using lotto systems.
        The book came as a result of long years of experience in creating and improving lottery systems. I have worked as a freelance writer for a number of lottery publications in Europe and North America, and I have spent many years in research in Combinatorics, an area of Mathematics, that among other things deals with combinatorial lottery systems (or coverings, as they call them in Combinatorics). I hold an M.Sc. and Ph.D. in Mathematics and work as a university professor. All of my publications, including both my master's thesis and doctoral dissertation, are related to lottery systems. The objects in this book originate from the most recent research in the area, and are based on expert knowledge of the subject. That is why this book has no match among the publications and software in existence.

Back to my playing experience: Indeed, I played, both individually and in pools, and I played enough to win: a couple of second tier prizes, a number of third tier prizes; I even hit the top tier once, a small jackpot, back in Europe, but I am still chasing the elusive Big Jackpot. The "small" jackpot that I won was actually a car. I never drove that car. I chose to receive the equivalent of the value of the car in money and continued playing until I hit a 5-win (in pick-6 lotto), first with another player and then another one, just by myself, winning multiple smaller prizes along the way. That was over 30 years ago; the cost of a ticket was 10 cents and it was quite affordable to play a system. I was a student at the time, young and excited about many things in life; I spent a good portion of my winnings on partying non-stop for months, and, of course, on some more lottery play. I believe I made many people happy to know me and made many new friends as well. Perhaps, I was luckier than average at that time; still, I did not become rich; after all, it was a car and some small to medium wins. Nevertheless, these were good size wins for the time and I had a lot of fun dealing every day with sums higher than my monthly allowance. Overall, I was happy to have had this experience, because it earned me a lifetime interest in a subject that later became my area of research and studies and kept me quite busy over the years. I had already been using lotto systems, but it was at that time that I really became interested in the subject, and I started thinking on how to improve the performance of these systems to the absolute maximum, by reducing the number of combinations for any given guarantee to the absolute minimum. I had a lot of success in this direction and I am still fascinated by the subject. Many users of lotto systems are enthusiastic about my book, others are cautious; I believe this is the case with every gambling book. I just offer my knowledge and expertise to the players and groups who are willing to use it. The book might help you win more prizes or win prizes more often. In any case, I am happy to offer you a new perspective on the lottery, new ways of enjoying the lottery as a game, and not just as the five minutes thrill of watching the draw on TV. I believe that playing lotto should be fun, just like any other game of chance, and I am glad that my book is a step in this direction.

Here is a bit more history in support of my credentials: Back in 1987, I started writing for a weekly newspaper on sport and lotteries, and then I started a column devoted to lottery systems in a second, similar, newspaper. I moved to Canada in 1993 and continued my contributions to these two newspapers for several years after that. After spending over 10 years in studies and research in Combinatorics, working mostly on coverings (the mathematical term for lottery systems), in 2001, I published the first edition of the book Combinatorial Lottery Systems (Wheels) with Guaranteed Wins, which quickly became the Amazon bestselling book on lotto systems for pick-6 lotteries. (A pick-6 lottery is one where the main draw has 6 numbers, plus possibly one or two bonus numbers.) A second edition of that book was published in 2010, a third one in 2012, and a fourth in 2020 (you can find information about the newest edition of the pick-6 book here). A pocket version called The Ultimate Guide to Lottery Systems was published in 2009 and translated into several languages including German, Spanish, Portuguese, and Japanese. Meanwhile, readers kept asking about a similar book for pick-5 lotteries, and I published such book in 2011; a second edition was produced in 2020; available at Amazon Kindle.

This book can be used for any pick-5 lottery in the world. It can be used in almost any English speaking country, as there are such lotteries in each of Canada, USA, United Kingdom, South Africa, and Australia. There are pick-5 lotteries in almost any country in Europe, and also in many other places, including Antigua, Argentina, Barbados, Dominican Republic, Ghana, Jamaica, Japan, India, Mexico, Philippines, Taiwan, Turkey, Trinidad and Tobago, to name a few. A pocket version of the pick-5 book has been produced (and also translated in French).

Some of my readers might wonder why I decided to write books on this subject. Well, I have been in the business of creating and publishing lotto systems for a long time, so that the idea of writing a book was not that foreign to me; nevertheless, I did not really intend to write books, not until I realized that there are several such books in existence and I saw them: they all shared the same characteristics: quite outdated and poorly written; often giving misleading information to the reader. They all exploited lotto systems from older, European sources, such as the Bulgarian Lottery Corporation Book, or a similar book from Hungary. Systems from these sources were recycled and published as new material by a number of authors. More recently, some of these authors updated their collections from Internet sources and published slightly more up-to-date editions; still the state of the art was out of reach for them and consequently for the lotto players. It was at that time that I decided to make some difference by writing new books not only for lotto players who were already familiar with using lotto systems but also for players who were willing to start experimenting with them. I was up to the challenge. Before I started writing, I carefully studied what other sources had to say about lotto systems (wheels). During my career as a professional mathematician and creator of lotto systems I collected every imaginable piece of literature and software on the subject. I also collected and studied pretty much everything written in the area of coverings (the mathematical term for lotto systems or wheels). I was quite happy to discover that many of my systems are better than the existing ones in terms of performance. At that time, I realized that I should start publishing my findings. In the process, I found many more new record breaking wheels and published them in various lotto publications. I was not in a hurry to publish books. I realized that the process of "getting the big picture" might take several years, and several years it took. Finally, in 2001, the first edition of my pick-6 book was published. It came out at a time when I was absolutely satisfied with the content and the quality of the product, at a time I considered its level to be well above the level of any competing product. The book quickly found its permanent place in the lotto products market and has been the number one bestselling book on lotto systems for a long time. The tradition of superior quality has been fully maintained in the subsequent editions of my pick-6 book (2010, 2012, 2020) as well as throughout the editions of my pick-5 book (2011, 2020).

 


 

 

 

Introduction

 

Lotteries

 

 

       Odds of being struck by lightning are better than the odds of winning a lottery.

 

Lottery Myth

 

        The number of possible draws in any lottery is huge and the probability of winning the jackpot is small. Still not as small as the probability of being struck by lightning, as the following excerpt from Duane Burke's address,  Top Ten Myths About Lottery (and Why They Are Not True) suggests: "Statistics gathered by NASPL (North American Association of State and Provincial Lotteries) indicate that in one year alone (1996) 1,136 people won a million dollars or more and an additional 4,520 won $100,000 or more by playing North American lotteries. By contrast, 91 people were killed by lightning during that same year in the United States." This was in year 1996, but the number of people struck by lightning probably remains about the same over the years, while many new lotteries are created every year, and therefore many more opportunities for winning a top prize arise. You would probably agree that the myth is busted. Indeed, the probability of winning the jackpot ranges from the relatively high 1/15,104 in a 5/20 lottery to the staggering low 1/302,575,350 in the multi-state double-pick lottery Mega Millions. Aside from winning a Jackpot, the players have other ways to win something with their tickets; in many lotteries, they can even enter a draw and win a prize with a non-winning lottery ticket, and, as Burke continues: "What kind of a second prize does a lightning strike offer?'' To be fair to the supporters of the myth though, I will finish this "lightning" diversion with a comment from a reader of Burke's article: "The truth is: There are way less people trying to be struck by lightning (than those trying to win a Jackpot). If you were really trying, your odds are pretty good!'' The chances of winning prizes smaller than the jackpot in any lottery are much better. That is why, in the quest for the elusive jackpots, many players prefer to play with a well-organized group of tickets, so that they can win guaranteed smaller prizes. Such a group of tickets is what we call a lottery system. Lottery systems can be thought of as an interesting and entertaining, but also very precise strategy for playing the lottery. Although the best lottery systems were created by using quite advanced mathematics and computations, they are very easy to use. In fact, to use lottery systems, all you need to know is how to count up to the number of balls used in your lottery. No mathematical knowledge is either assumed or required for playing a lotto system. In this book, I will explain what a lottery system is and how to use it, and I will also provide you with a number of excellent lotto systems to diversify your lotto experience. A lottery is a game. Games are entertaining. I hope this book will show you, among other things, how to extract more entertainment value from your lottery playing experience.

We start with a brief discussion on lotteries. A pick-5 lottery is a game where one has to correctly guess several (usually 2 or more) of the numbers (5 in our case) drawn from a larger set, of, say, 39 numbers, for example. In various places of the world, there are lotteries with 20, 25, 30, 31, 32, etc., up to 90 numbers. The size of the lottery often depends on the size of the country (or state) where the lottery is played. Some lotteries introduce bonus (supplementary) numbers, which might be drawn from the same set as the main 5 numbers (such as Maryland's Bonus Match 5, Road Island Wild Money, New Mexico Roadrunner Cash, etc.), or from a different set (Powerball, Mega Millions, EuroMillions, Loto, Thunderball, California SuperLOTTO Plus, Hot Lotto, Megabucks Plus, Wild Card2, Kansas Cash Lottery, Tennessee Cash, etc.). You can use the lottery systems presented in this book for ANY such lottery in the world, even for lotteries that might be introduced in the future and have a number of balls outside the range 20-90.

In this book we focus on 5-numbers-drawn lotteries, but most of the existing lotteries operate on the same principles as the 5-numbers-drawn lotteries. If the main draw consists of 5 numbers, the lottery is usually referred to as a pick-5 lottery. If the main draw consists of 6 numbers, the lottery is called a pick-6 lottery. There are also pick-3, pick-4, pick-7 and even pick-more lotteries (Keno, for example). In any lottery, players fill out slips containing one or more tables, each containing all of the numbers of the lottery. For example, in a 5/39 lottery, each table has all 39 numbers, and the player fills 5 numbers in each table. We also refer to these tables as tickets. (Other sources use terms such as games, plays, or boards.) A combination is the set of 5 numbers filled in a particular table. When the draw comes, 5 numbers are drawn from the set of all numbers in the lottery. Players win whenever they correctly guess usually 2 or more of the numbers drawn in one of their tickets. The number of tables on a playing slip does not really matter for the application of our lottery systems.

 

 

 

What Is a Lottery System?

 

 

If you watch morning shows or check your lotto publication regularly, you have probably seen it already, a syndicate of 7, or 10, or any number of people, won the big jackpot; or a smiling individual holding the big check and sharing his/her joy with the world. You have dreamed of you being there on the picture, right? I have dreamed about that too; still do, as a matter of fact. Occasionally, the winners will mention how they got there, some of them will praise their lucky numbers, and others will claim they used a lottery system. Yet others may have used a lottery system, but they will shy away from mentioning it to the public, they will just enjoy the money and try to win it again. So, what is the hype about? It is about the fact that if you want to play more numbers and you want to have certain guaranteed wins, then you have to use a lottery system (or a lottery wheel, as some authors call it) to organize (or wheel) your numbers. What is a lottery system and how can it help you diversify your lottery playing experience? In what follows, the lottery systems will be demystified for you, and you will be provided with a nice collection of great lotto systems.

Lottery systems are sought and used by lottery players throughout the world. Lottery systems are considered to be not only entertaining, but also a very well-organized way of playing the lottery. This book introduces the best-known systems (also called trapping systems, or wheeling systems, or wheels in short; mathematicians call these objects coverings). I will talk more about what "best" means in the next two sections. The process of using a lottery system (or wheel) is sometimes referred to as wheeling. There is a reason for that which originates from the construction of some systems; I will talk more about it later, in part IV of the book. Lottery players like using a system, because a system guarantees wins in the same way a single ticket does, while it allows playing with many numbers (6,7,8,9,10,11, etc.). I believe that using a lottery system from this book is more entertaining than just using a system from other sources or playing a random collection of tickets, and one of the reasons is the following: The possible wins for each system can be studied in advance from the Table of Possible Wins, a feature that has been fully developed and implemented for the first time ever in my books. Players like the fact that playing with systems provides a steadier stream of wins compared to playing with a random collection of tickets. I will expand a bit on that in the next section. Players also like the fact that playing with a system increases the chances of winning. This comes with a price, of course: The price of purchasing more than just one ticket. However, you might have already been playing with more than one ticket per draw, or, perhaps, you have considered doing so. Then this book is exactly the book you need.

Suppose for example, you want to play with 9 numbers instead of just 5. Naturally, you are willing to play more than one ticket and you want a certain guarantee. Let us say that you want a 3-win whenever 3 of the numbers drawn are in your set of 9 numbers. This means you want a set of combinations (tickets) that covers every possible triple out of your 9 numbers. Such a set of combinations is an example of a lottery system. You can find a lottery system with this property under #26 in the book. In this example, the lottery system has the property that any triple out of your set of 9 numbers is contained in at least one of your combinations, so you get a guaranteed 3-win whenever 3 of your 9 numbers are drawn.

In a sense, a lottery system expands the guarantee that you have on playing just 5 numbers (one ticket). If you just play one ticket and correctly guess 3 numbers, then you are guaranteed a 3-win. The lottery system in our example has the same guarantee, a 3-win if 3 numbers are guessed correctly, except that now you play a larger selection of numbers; 9 in this case. (In this book we also present systems with 3 if 3 guarantee for playing anywhere between 7 and 27 numbers.)

 

 

 

Guarantees

 

 

 All of the lottery systems presented in this book have certain guarantees. In this section, I will explain what the advantages of using a lottery system are and what exactly a guarantee means when you play a lottery system. Using a lottery system assumes that you play with more than five numbers, and you want to organize your numbers in such a way that a certain minimum win is guaranteed. If you play with just one ticket and you correctly guess 3 numbers, then you are guaranteed exactly one 3-win. A possible question was mentioned earlier, namely: How to build a system that will guarantee a 3-win on 3 guessed numbers, if you play with not 5, but say, 9 numbers. Let us look at our previously mentioned example from the book, System #26. By using System #26, you are guaranteed a 3-win whenever 3 of your 9 numbers are among the 5 numbers drawn. The system shows that you only need 12 combinations (or 12 tickets) to do so. Out of 9 numbers, one can form 126 distinct quintuples (5-number combinations); the non-mathematicians can take my word for it; those with some mathematical preparation can apply the formula C(9,5)=126. Therefore, you only need to play 12 of these 126 combinations to achieve the 3 if 3 guarantee. What are the advantages of playing for such a guaranteed win? Let us compare playing with 12 random tickets against 12 tickets chosen according to our lottery system. We should mention that the probability of a 5-win is the same for each ticket. However, if any 3 of the numbers drawn are among the 9 numbers chosen by you, then the 12 tickets of the lottery system guarantee at least one 3-win, while 12 random tickets (on the same 9 numbers) guarantee nothing! This illustrates one of the main advantages of using a lotto system. Suppose you played your 9 numbers in the 12 combinations of System #26 for a long time and hit 3 of the numbers drawn, say, 10 times, over this period of time. Every time you hit 3 of the numbers drawn, you won at least one 3-prize. Had you played your 9 numbers in 12 random combinations, you would have probably hit a 3-win several times, but, most likely, not all 10 times, as with the system. In a worse case scenario, you could have missed the 3-win in each of the 10 draws in which you hit 3 of the numbers drawn! The reason is quite simple: Random combinations do not come with a guarantee; a system does!

If you want to follow everything mentioned in the last two sections, you can check our companion site www.lottowheeling.com, where you will find an interactive example of System #26, which we use as an illustration here. It allows you to substitute your numbers, generate your combinations, check how the guarantee of the system works, and also check it against any possible draw if you enter the numbers drawn; the same functionality is available for every system presented in this book.

 

 

 

Minimality

 

Lottery systems have been used by lottery players throughout the world. In fact, some European lottery corporations have integrated lottery systems in the automated processing of lottery tickets. Most of the existing implementations are based on out-of-date systems. Lottery players have always been attracted to the most economical lottery systems, that is, systems achieving a certain guarantee in the minimum possible (or known) number of tickets. Recent advances in the research on coverings made it possible to reduce the number of combinations in many systems to the absolute minimum in the entire range of possible guarantees. These advances are reflected in my book.

Minimality is an important quality of all lottery systems in this book. The logic behind seeking minimal systems is simple: Let us assume you play with 9 numbers. If you could get a guaranteed 3-win by using a system in 12 combinations (system #26 from this book) and a system in 20 combinations, then which one would you choose? The answer is obvious: You would most likely prefer to get the guaranteed 3-win in the fewest number of tickets possible, in this case, in 12 tickets. Some players might argue here: OK, but if I play 20 tickets, I will have 20 shots at a 5-win rather than just 12. True, but if you really want to play 20 tickets, perhaps, you can play for a higher guaranteed prize, or you can play more than 9 numbers for the same guarantee, thereby giving yourself a better chance to capture all of the drawn numbers in your larger set. For example, in 20 combinations, you could play system #28, which has 11 numbers and still guarantees a 3-win if 3 of your numbers are drawn, or you could even go down to 18 combinations and play system #108, which has 9 numbers and guarantees not one, but two 3-wins if 3 of your numbers are drawn! (Yes, you read it right, you do not have to double the number of combinations from 12 to 24 to get the double guarantee! More on that can be found in Part II of the book.) So, the minimality concerns the following question: How many tickets have to be played in order to have a certain guarantee? Clearly, you want to achieve this guarantee in the minimum number of tickets possible. Well, you are at the right place: All of the systems presented in this book use the minimum known number of tickets. The systems are combinatorial objects that have been extensively studied by mathematicians and computer scientists. The systems in this book are either impossible or very unlikely to ever be improved. Some of the systems represent classical results in combinatorics; others originate from recent research. Many of the systems have been obtained by the author and described in depth in a series of scientific papers. Others have been obtained via hundreds of hours of programming and computations. As a result, all of the systems are currently the best (in the minimum number of tickets) known. For most of the systems presented here, we can say even more: They are mathematically minimal, meaning that no further improvement (that is, reducing the number of tickets while preserving the guarantee) can ever be done.

Let us look one more time at our example, System #26. It has been proven that the minimum number of tickets is 12 for the given guarantee. In other words, if you want to play 9 numbers and you want a guaranteed 3-win if 3 of your 9 numbers are drawn, then you need to play at least 12 tickets. System #26 achieves the 3 if 3 guarantee in exactly 12 tickets, and this is the minimum possible number of tickets for that guarantee. That is why we call such a system mathematically minimal.

 

 

 

Tables Of Wins

 

 A new feature and a very important quality of this book is the full table of possible wins for each presented system. The tables give the distribution of wins in all possible draws and are excellent tools for determining which system to choose and what could the expected win be once you hit some (usually at least two) of the numbers drawn. Let us take a look at the full table of possible wins for System #26. It is somewhat long. This explains why for some systems I present an abbreviated table of wins. Here is the full table first:

 

Guessed

5

 4

  3

  2

  %

  5

 1

 -

 8

 2

 0.79

 

 1

 -

 7

 4

 6.35

 

 1

 -

 6

 4

 0.79

 

 1

 -

 5

 6

 1.59

 

 -

 3

 6

 2

 4.76

 

 -

 3

 5

 3

 1.59

 

 -

 3

 4

 5

 12.70

 

 -

 3

 3

 6

 6.35

 

 -

 2

 7

 2

 4.76

 

 -

 2

 6

 4

 9.52

 

 -

 2

 6

 3

 17.47

 

 -

 2

 5

 5

 9.52

 

 -

 2

 5

 4

 6.35

 

 -

 2

 4

 6

 3.17

 

 -

 1

 8

 2

 9.53

 

 -

 1

 7

 4

 3.17

 

 -

 -

 10

 1

 1.59

  4

 -

 1

 4

 6

 0.79

 

 -

 1

 4

 5

 6.35

 

 -

 1

 3

 6

 17.46

 

 -

 1

 3

 5

 1.59

 

 -

 1

 2

 8

 10.32

 

 -

 1

 2

 7

 4.76

 

 -

 1

 2

 6

 4.76

 

 -

 1

 1

 10

 1.59

 

 -

 -

 6

 5

 1.59

 

 -

 -

 6

 4

 3.17

 

 -

 -

 6

 3

 6.35

 

 -

 -

 5

 5

 9.52

 

 -

 -

 5

 4

 12.70

 

 -

 -

 4

 7

 9.52

 

 -

 -

 4

 6

 9.53

  3

 -

 -

 3

 3

 2.38

 

 -

 -

 3

 2

 1.19

 

 -

 -

 3

 -

 1.19

 

 -

 -

 2

 6

 9.52

 

 -

 -

 2

 5

 7.14

 

 -

 -

 2

 4

 16.67

 

 -

 -

 1

 8

 4.76

 

 -

 -

 1

 7

 21.43

 

 -

 -

 1

 6

 35.72

  2

 -

 -

 -

 5

 2.78

 

 -

 -

 -

 4

 27.78

 

 -

 -

 -

 3

 69.44

 

   Each line represents a possible distribution of wins. The last column shows the probability of the corresponding distribution of wins. If you play System # 26 and 3 of your 9 numbers are drawn, then the system guarantees you at least one 3-win. Note that you will actually win more than that: There are nine possibilities that are clearly seen from the section of the table corresponding to  guessed numbers; these are given in the nine lines of that section. We note that the possibilities with the same number of 3-wins are not essentially different, so we can actually group them together. You can now check that you either get one 3-win plus 6-8 2-wins and this happens in 4.76+21.43+35.72=61.91% of the cases (or, we say it happens with probability 61.91%), or you get two 3-wins plus 4-6 2-wins (probability 9.52+7.14+16.67=33.33%), or three 3-wins plus 0-3 2-wins with probability 2.38+1.19+1.19=4.76%. Similarly, if you look at the section on  guessed numbers, you will see that if 4 of the numbers drawn are among your chosen 9, then you will either get a 4-win plus one, two, three, or four 3-wins with probability 1.59%, 19.84%, 19.05%, or 7.14%, respectively; or, four, five, or six 3-wins with probability 19.05%, 22.22%, or 11.11%, respectively. In each of these cases you also get 2-wins (anywhere between 3 and 10) The percentages add up to 100% in each of the four sections of the table.

Presenting individually every line in the full table is not always physically possible; in many cases the full table would contain several hundred entries. In such cases, and in order to keep the book compact, I present an abbreviated table. In fact, you will notice that the table of possible wins for System #26 looks a bit different from the full table presented earlier in this section; we give the abbreviated table here as well, for easy comparison.

 

Guessed

 5

 4

  3

  2

  %  

  5

 1

 -

 5-8

 2-6

 9.52

 

 -

 3

 3-6

 2-6

25.40

 

 -

 2

 4-7

 2-6

50.79

 

 -

 1

 7-8

 2-4

12.70

 

 -

 -

 10

 1

 1.59

  4

 -

 1

 4

 5-6

 7.14

 

 -

 1

 3

 5-6

19.05

 

 -

 1

 2

 6-8

19.84

 

 -

 1

 1

 10

 1.59

 

 -

 -

 6

 3-5

11.11

 

 -

 -

 5

 4-5

22.22

 

 -

 -

 4

 6-7

19.05

  3

 -

 -

 3

 0-3

 4.76

 

 -

 -

 2

 4-6

33.33

 

 -

 -

 1

 6-8

61.91

  2

 -

 -

 -

 5

 2.78

 

 -

 -

 -

 4

27.78

 

 -

 -

 -

 3

69.44

 

  Note that no essential information has been lost: An abbreviated table still comprises all possible distributions of wins, but most of the lines in such a table actually represent several lines from the full table. We can justify such abbreviations by observing that what matters most are the highest-ranked prizes. The number of such prizes is given by the first number of a line in the table. What one gets in lower-ranked prizes is generally a small part of the entire win. In abbreviating the full tables I tried to keep distinction between the highest prizes. In other words, I tried to start every line in a table of possible wins by a single number rather than by an entry of the range type a-b. For example, I could have further abbreviated the second section (corresponding to 4 of your 9 numbers drawn) of System #26 as

 

Guessed

 5

 4

  3

  2

  %  

  4

 -

 1

 4

 5-6

 7.14

 

 -

 1

 3

 5-6

19.05

 

 -

 1

 2

 6-8

19.84

 

 -

 1

 1

 10

 1.59

 

 -

 -

 4-6

 3-7

52.38

 

 

but I did not, because the last line of the section would not have been quite informative. If that section was much longer, perhaps, an abbreviation like

 

Guessed

 5

 4

  3

  2

  %  

  4

 -

 1

1-4

5-10

47.62

 

 -

 -

 6

 3-5

11.11

 

 -

 -

 5

 4-5

22.22

 

 -

 -

 4

 6-7

19.05

 

 

would have been justified, as every line still starts by a single number entry rather than one of the range type a-b.

 

We include a further brief explanation for those readers who are not familiar with probabilities: Let us look again at the section of the table corresponding to hitting three of the numbers drawn. Suppose you have played long enough with the same system and hit 3 of the numbers many times. Then, in approximately 62 out of 100 cases (61.91, to be precise), you will have one 3-win, in 33 (33.33, to be precise) out of 100 cases, you will have two 3-wins, and in 5 (4.76, to be precise) out of 100 cases, you will get three 3-wins.

 

 

How To Use The Systems

 

  Playing with the systems is very easy: In our example (System #26), where you play with 9 numbers, you will just have to perform the following simple operations:

 

 (1) write the numbers from 1 to 9 in a line (these are the numbers in the original system);

 

 (2) write your 9 numbers in a line below the first one;

 

 (3) substitute each number in the original system with the corresponding number from the second line to obtain your set of tickets;

 

 (4) fill your combinations in the playing slips.

 

 For example, if your 9 numbers are 2,3,7,12,15,18,19,22 and 24, then you will have the following.

 

 Numbers in the original system:

1

2

3

4

5

6

7

8

9

Your numbers:

2

3

7

12

15

18

19

22

24

 

 

 

  Original system:

 

  Your set of tickets:

1.

 1

 2

 3

 7

 8

 

1.

2

3

7

19

22

2.

 1

 2

 4

 7

 9

 

2.

2

3

12

19

24

3.

 1

 2

 5

 6

 7

 

3.

2

3

15

18

19

4.

 1

 3

 4

 5

 6

 

4.

2

7

12

15

18

5.

 1

 3

 4

 8

 9

 

5.

2

7

12

22

24

6.

 1

 5

 6

 8

 9

 

6.

2

15

18

22

24

7.

 2

 3

 4

 5

 8

 

7.

3

7

12

15

22

8.

 2

 3

 5

 7

 9

 

8.

3

7

15

19

24

9.

 2

 3

 6

 8

 9

 

9.

3

7

18

22

24

10.

 2

 4

 6

 7

 8

 

10.

3

12

18

19

22

11.

 3

 4

 6

 7

 9

 

11.

7

12

18

19

24

12.

 4

 5

 7

 8

 9

 

12.

12

15

19

22

24

 

You can write your numbers in any order in the second line, not just in increasing order; still the guarantee of the system will be the same; moreover, the table of possible wins will be the same (however, you might hit a different line of the corresponding section of the table of wins). Usually the systems are balanced, in the sense that all numbers are almost equally represented. That is why I recommend arranging your numbers in increasing order; then the substitution can be made in the easiest possible way. Of course, if the system is not completely balanced, then you might choose to put your favorite numbers under the system numbers with the highest number of occurrences.

Let us illustrate one more time the guarantee of the system from our example (a 3-win if 3 of your numbers are drawn): Suppose the numbers 7,12 and 24 are drawn, then the system must bring you at least one 3-win. Indeed, it is easy to check that this is so. In fact, you will get two 3-wins (in tickets 5 and 11).

The book also contains systems with different types of guarantees, for example, a guaranteed 3-win if 4 of the numbers drawn are guessed correctly; or a guaranteed 4-win if 5 of the numbers drawn are in your chosen set of numbers. The use of such systems is the same as in the example above. We also introduce systems that have never appeared in the lottery literature or software: Systems where the main guarantee is multiple prizes, for example, a guarantee of two 3-wins if 3 of the numbers drawn are in your set of chosen numbers, or a guarantee of three 4-wins if 5 of the numbers drawn are in it. Again, these systems can be used in the same way as explained at the beginning of this section.

Finally, I want to bring to your attention a question that players ask sometimes: I have used a system and won, but my set of prizes can be found nowhere in the table of wins. What is wrong? There are two possible explanations: You have either produced your set of tickets incorrectly, or you have made a mistake while filling your playing slips. So, do the substitution carefully, avoid any distraction while filling the playing slips. A single mistake can cost you a huge prize...

 For those of you (or your syndicate) who play a larger system, or for anyone who wants to save the time needed for substituting your numbers, generating all the tickets, and even checking the tickets against the drawn numbers, we are offering a service/tool that will perform operations 1)-3) electronically, on the web; it will also check your tickets against the drawn numbers and show your wins. Another service will perform operation 4) for some lotteries. Check our companion site www.lottowheeling.com for the added functionality.

 

Choosing Your Numbers

 

In order to get your guaranteed wins you still have to guess correctly some of the numbers drawn. A natural question is then how to choose the right numbers. Lottery officials tell us that any draw is totally random and unbiased. Mathematicians and statisticians are consulted to make sure that any particular draw represents a completely random selection of numbers. I agree that lottery officials make any effort to secure a fair game for all players. I trust that and I trust that my fellow mathematicians and statisticians do their best to make the lotteries as unbiased as possible. Still, any draw is a physical process; even generating random numbers by a computer is a physical process, and the numbers generated are as random as the skills of whoever programmed and patented the random generator. There are not many types of physical machines that draw the numbers. Perhaps the most popular mechanical solution is a machine that mixes and draws the numbers by using compressed air. The numbers are painted or imprinted on balls. The balls are usually made of light plastic; often they use table tennis balls. Let us assume, for the purpose of this discussion, that the lottery has 39 numbers. Initially, the balls are arranged in some random manner in a container. When the draw starts, the balls get pushed by the compressed air into a transparent sphere where they bounce on the inside surface of the sphere under the action of the compressed air and get mixed well, until one of them is captured in a pipe, through which it rolls into a display. The process continues until all five numbers are drawn (6 or 7, if there are bonus numbers in the lottery). Now, this is a physical process all throughout. There are so many factors that affect the draw that one might easily be tempted to question the integrity of the entire process: The initial arrangement of the balls, the weight and shape of the balls, the air pressure provided by the compressor, etc. One question which is frequently discussed at lottery forums, and which I do not have to answer for you here is the following: You paint the numbers 1 and 34 on two identical balls; are they still identical? I know lottery officials do their best to ensure the lotteries are unbiased. They are happy if the public believes their lottery is unbiased. The reality is: The public chooses what to believe in. Some people do not trust random generators, for example. Random generators were proven to be not that random in the past. You might have heard stories about lottery corporations that decided to draw the numbers by computer and got flooded by letters from players questioning the integrity of the draw. Mechanical ways to draw the numbers are more trusted: One sees what is going on; testing looks easier, etc. Now, we have to very well realize that lottery officials do as they say; they are truly interested in the complete randomness of their lotteries. After all, why would they need to cheat the public if the lottery usually takes approximately 50% of all money that goes in the game? Of this 50%, probably about 5-10% goes to expenses associated with running the lottery, the rest goes to various charities, so money spent on the lottery actually serves good causes, if this makes you feel good. Lottery is a game of chance, it is a gambling game and as such it is no different than any other gambling game. The house (the lottery officials in this case) is truly interested to provide a fair game.

Still, gamers have exploited `random' processes to the tune of making millions by tracing bias in such processes. They have exploited pretty much everything: From gaming terminals in pubs and casinos, to roulette tables in the big gambling cities, to poker and other gaming sites on the Internet. They have exploited bias in physical devices such as roulette tables, but they have also exploited computer games, such as online poker where the cards are dealt based on numbers between 1 and 52 generated by a random number generator (RNG). Stories like that are all over the Internet and the available literature. So, you decide for yourself what you believe or not believe in when you play the lottery.

 

... … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

(read the full text of this section in the book)

 

 

Can You Win The Top Prize in Your Lottery by Playing With a System?

 

 Of course, you can! In fact, every ticket of your system can win the top prize. Here we assume that the top prize in your lottery is a 5-win. This is not exactly the case in the Mega Lotteries such as Powerball, Mega Millions Thunderball, and Euro Millions, although you still have to hit all 5 of the main numbers drawn; there you also have to hit one or two bonus numbers; these are essentially two-pick lotteries and will be discussed separately. The advantage of using a lottery system is that even if you do not win the top prize, you will still get your guaranteed wins provided the draw hits your numbers. That is one of the reasons why lottery players and groups of players prefer to use systems. A lottery system gives you the opportunity to chase the top prize in an organized and entertaining way, and also guarantees some smaller prizes if less than 5 of your numbers are drawn. Of course, if you hit the top prize by using a lottery system, then you will also win a number of smaller prizes.

A system that guarantees the top prize (that is, a 5-win if 5 of your numbers are drawn) is expensive. Below is a table that gives you the number of combinations you need to play in order to get every possible combination out of your numbers.

 

Numbers played

Number of combinations

6

6

7

21

8

56

9

126

10

252

11

462

12

792

13

1,287

14

2,002

15

3,003

16

4,368

17

6,188

18

8,568

19

11,628

20

15,504

21

20,349

22

26,334

23

33,649

24

42,504

25

53,130

26

65,780

27

80,730

28

98,280

29

118,755

30

142,506

 

 The number of tickets increases very fast. A system that contains all possible combinations is called a complete system. If you play with n  numbers in a pick-5 simple lottery (no bonus numbers), then the number of combinations in a complete system can be computed by the formula [n(n-1)(n-2)(n-3)(n-4)]/120. Complete systems are not included in this book. Many lotteries allow using such systems automatically and provide the corresponding tables of wins.
        Now, how difficult is to be a big winner? The Jackpots vary considerably in type and size. There are lotteries with a fixed Jackpot, which can be any sum starting as low as $20,000. In this type of lottery, there is usually a restriction on the number of Jackpots that can be given at the fixed value in any given draw. For example, if the fixed value of a Jackpot is $100,000, the lottery might stipulate that they pay up to three such Jackpots (up to $300,000 designated for first tier prizes) in any particular draw. If there are five Jackpot winners, the value of the Jackpots will be reduced to $60,000. There are lotteries where the designated Jackpot sum rolls over to the next draw, essentially making the Jackpot bigger and bigger until a player or several players win it. This is the case with the big lotteries such as Powerball, Mega Millions and EuroMillions, as well as some big state lotteries such as California SuperLOTTO Plus. There are also lotteries where the designated Jackpot sum goes to lower tier prizes (rolls down) if no one hits the Jackpot. These lotteries are particularly well suited for system players, because the expected value of the lower tier prizes is higher than in other lotteries. Some of the roll over lotteries have restrictions on the number of roll overs (or the size of the Jackpot, as in EuroMillions, where the current limit is 185,000,000 euros) before the entire sum rolls down to lower tier prizes, or stays at a fixed limit and the extra money goes to lower tier prizes if there is no jackpot winner. Winning a Jackpot is the ultimate goal of any player and it is always good if it happens, but, it might not even make you a really big winner, while in some cases it might make you insanely rich. Some lotteries are so big that even winning a non-Jackpot prize could make you a big winner. So it is difficult to evaluate the chances of a big win over a lifetime of playing. In what follows, I will just give a simplified argument, an approximation to the reality to illustrate how difficult it is to win a life-changing sum in a lottery. I am now thinking how difficult it is to actually precisely define the meaning of a "life-changing sum''; obviously it depends on the individual player and his/her financial status.

Let us assume that the lottery is a fair game of chance. Let us assume that a ticket costs a dollar and you (or your syndicate) play 10 tickets 100 times per year (this will approximately be the case if your lottery has two draws per week). This means you (or your group) spend about a $1,000 per year on lotto tickets. Now, to win a $1,000,000 jackpot, you have to be ready to wait some period of time in the range of 2,000 years (recall that the lottery returns just about 50% of the money that goes into it). The reality is: It might not happen to you (or your syndicate). Now, let us say, you play with the same sum for 40 years. The chance that you will win the one million jackpot during that time will be 40/2,000=1/50 or just about 2%. In other words, for one big win over a lifetime of playing, there must be 49 small losers who will pay for it. When I say "small" I mean it relatively; there will be many small wins that will partially cover the cost of playing for the small losers; one could end up being a winner even without winning big jackpots along the way, hitting, say a top prize in a small pick-5 lottery, or several 4-wins over the years, or hitting a 5-win or several (1+4)-wins in one of the Mega Lotteries. As I mentioned before, the above computation is quite an approximation to what could happen in reality. There are several factors that will have to be taken into account if we want to precisely evaluate how difficult it is to win a jackpot and how long the "waiting period" will be for it to happen on average, given that you play a fixed amount each draw. A precise computation is impossible, due to these additional factors. First of all, we did not factor the small wins which will come along the way and which you will probably put in the play to partially finance your game. The precise computations will also depend on how your lottery distributes the money for the different tier prizes. Another, very important factor will be the size of the jackpot. In fact, much larger than $1,000,000 jackpots happen on a regular basis, and although this does not significantly affect the "waiting period", it certainly affects the reward if it does happen. In any case, the point to keep in mind is: It is difficult to be a Jackpot winner. Well, why do we play the lottery then? First of all, there are supposedly factors that can change the numbers in favor of a smart player, or so we believe; I spoke about these in the section Choosing Your Numbers. Also, it is the "one never knows" factor, or the "it could be me" factor; the same factors that motivate people in any form of gambling. This is a good place for the usual warning, which is valid for all forms of gambling: Do not play with money that you cannot afford to lose! Playing the lottery is often compared to a high risk investment: The risk to lose is high indeed, but the return is immense if you beat the odds.

 

 

The Odds Of Winning

 

The odds (or probability, or chance) of hitting the top prize of your lottery are the same for any particular ticket. A system contains several tickets, so the chance of winning the top prize increases with the number of tickets played, but so do the expenses. What a system really does is that it guarantees smaller wins, while just a random selection of tickets usually guarantees nothing, even if all of the 5 numbers drawn are in your set of numbers (assuming that you play with more than 8 numbers). The chance of winning any particular set of prizes is clearly seen from the tables of wins.

 

 

Bonus Number(s)

 

Some lotteries introduce one or more additional numbers (we will call them bonus numbers) drawn either from the same set as the main 5 numbers, or from a different set, and pay prizes if you hit some of the main numbers plus one or two of the bonus numbers. For example, if you hit, say, 4 of the main numbers and the bonus number (or one of the bonus numbers in EuroMillions) then you will have a 4+1-win. Other winning combinations are also possible. All of our systems are valid for these lotteries. Due to the diversity in using bonus numbers, and in order to keep the book compact, I have chosen not to include information on bonus number prizes in the tables of wins, however, I will discuss a couple of additional strategies on using systems in some of these lotteries in the next two sections. If your lottery has a bonus number (or two bonus numbers, as in EuroMillions), you only need to double-check each winning ticket containing a bonus number (or two bonus numbers).

 

 

Playing Systems in the Big Double-Pick Lotteries: Wheeling the Bonus Number(s)

 

If you play one of the big double-pick lotteries and you want to use a system with just one bonus number chosen for all of the combinations of the system, then you can skip both this section and the next one. These are for players who want to play a system and want to include more than one bonus number, thereby creating a bigger system (a multiple of the chosen one) with guarantees such as a 3+1-win, 3+2-win, 4+1-win, etc.

Can we really wheel the bonus numbers? In some cases, we can. I will start with recalling that there are three types of pick-5 lotteries, ordered in terms of increasing complexity:

 

1.  Simple pick-5 lotteries (such as 5/32, 5/39, etc.)

2.  Pick-5 lotteries where the 5 main numbers and the bonus number are drawn from the same set (such as MD Bonus Match 5, RI Wild Money, NM Roadrunner Cash, etc.)

3.  Double-pick lotteries, where the 5 main numbers are drawn from one set and the bonus number(s) from another set (such as Powerball, Mega Millions, EuroMillions, Thunderball, Loto, California SuperLOTTO Plus, Hot Lotto, Megabucks Plus, Wild Card2, Kansas Cash Lottery, Tennessee Cash, etc.)

 

As I mentioned earlier, all of the systems from this book can be used in each of the three types of lotteries. For the lotteries of type 2, we cannot do much about the bonus number: It is drawn after the 5 numbers of the main draw from the same set, so we do not have a way to distinguish it from the other numbers or indicate it in the playing slips; therefore, we can only double-check the tickets where we have 2-, 3-, 4-, or 5-wins to see if we actually have 2+1-, 3+1-, 4+1-, or 5+1-wins. However, we can do more in all of the lotteries of type 3: We can actually wheel several bonus numbers, because the bonus numbers are chosen from a different set. The bonus numbers have different names in different lotteries: Mega Ball or Megaball (in CA SuperLOTTO Plus, Megabucks Plus and Mega Millions), Powerball number or Red Ball (in Powerball), Lucky Star Numbers (in EuroMillions), Thunderball (in UK's Thunderball), Chance Ball (in France's Loto), Hot Number (in Hot Lotto), Wild Card (in Wild Card2), Cash Ball (in Kanzas Cash), etc. In all of the type 3 lotteries (except for EuroMillions), 5 numbers are drawn from one set and the bonus number is drawn from another set. The difference in EuroMillions is that two bonus numbers (the Lucky Star Numbers) are drawn from the second set, rather than just one. Therefore, I will discuss EuroMillions separately from the rest of type 3 lotteries. This will be done in the next section.

 

... … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

(read the full text of this section in the book)

 

 

Wheeling Bonus Numbers in Euro Millions

 

... … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

(read the full text of this section in the book)

 

 

Finding Your Way Through the Book

 

The two tables presented at the end of this section give you a quick and easy way to compare the systems in the book and choose the most appropriate one in terms of either how many numbers or how many combinations you want to play. The systems are listed by guarantees in the Contents and numbered consecutively. The first table here lists the systems by the number of combinations in increasing order. It can be used to find a system in a preferred price range. The second table can help if you already have a clear idea on how many numbers you want to play.

The contents and both tables can be used to compare the systems in the book by various characteristics. More facts about the properties of the systems can be found by comparison of the tables of wins. Each system is presented as a separate article. The title shows the main guarantee of the system. Many systems have additional guarantees that are explained in the presentation. I have also included some comments featuring particular strengths of the systems, information on the balance of the system (number of appearances of each number in all combinations of the system), and information on minimality.

Many systems in this book have the maximum coverage property. I use it as a synonym of combinations being maximally apart, or as apart as possible, or combinations being maximally different. What does this property mean and why is it good for a system to have it? In order to explain it without using too much mathematics, I will just continue with an example. Let us focus on a particular system, say, the last system in the book, #126. It has 11 numbers, 11 combinations and in my comments I say that the combinations of the system are maximally `apart': Every two combinations differ in at least three numbers. This also means that every two combinations have at most two numbers in common. It also means that if we take any two combinations of the system, they only cover distinct triples (or distinct 3-wins, if we use the usual terminology). Each combination covers exactly 10 triples, so the entire system covers 11(10)=110 distinct triples. Any other system with 11 combinations (or just a set of 11 random combinations) which does not have the maximum coverage property will actually cover a total of less than 110 distinct triples (3-wins). Just for simplicity, let us assume that the system has only two combinations; take the first two combinations of System #126,

 

1

2

5

6

8

1

2

9

10

11

 

  The triples covered by these two combinations are all distinct, for a total of 20 triples. Compare to the following two combinations

 

1

2

5

6

8

1

2

5

9

10

 

They do have a triple in common (1 2 5), so the total number of triples covered is 19. If we allow combinations which are even less apart, we will get more repeated coverage of triples and the total number of triples will further go down. For example, the next two combinations have four numbers in common

 

1

2

5

6

8

1

2

5

6

9

 

and the total number of triples covered goes down to 16 (because all of the four triples contained in the common part 1 2 5 6 are repeated in both combinations). The same type of argument can be applied when we consider the coverage of pairs or quadruples. There is a formula which tells us how apart the combinations of a system can be (depending on the numbers played and the number of combinations), but the complexity prevents me from discussing it here. I will just say that whenever I claim good covering properties, it is based on strict mathematics.

Finally, in my comments to each system, I also discuss balance. My ranking of possible balance is as follows: A system is well balanced if each of its numbers appears in either n or n+1 combinations. A system is highly balanced if each of its numbers appears in the same number of combinations, and it is exceptionally highly balanced, if, in addition, every pair of numbers appears together in the same number of combinations. Even higher levels of balance are present in some systems (such as "every triple of numbers appears together in the same number of combinations"). 

One of the advantages of playing a highly balanced system is convenience: You do not have to rank your numbers according to how likely you think each number is to be drawn and then place numbers with higher expectation to be drawn under the numbers with higher occurrences in the system. A highly balanced system usually has other nice properties (symmetries) such as the maximum coverage property discussed above and also a shorter table of wins. 

 

Navigation Table by Number of Combinations

  

  

# of

 Num-

 Syst.

 # of

 Num-

 Syst.

 # of

 Num-

 Syst.

comb.

 bers

 #

 comb.

 bers

 #

 comb.

 bers

 #

 3

 7

 11

  36

 11

 110

  132

 12

 123

 5

 6

 1

  37

 17

 53

  132

 16

 20

 5

 7

 24

  37

 21

 74

  133

 20

 37

 5

 8

 12

  40

 22

 75

  133

 33

 86

 5

 9

 45

  43

 14

 31

  136

 17

 116

 5

 11

 64

  43

 18

 54

  136

 34

 87

 6

 12

 65

  48

 12

 111

  137

 26

 62

 7

 10

 46

  48

 13

 17

  151

 21

 38

 8

 8

 25

  49

 23

 76

  151

 27

 63

 8

 13

 66

  51

 10

 5

  157

 13

 8

 9

 7

 2

  52

 19

 55

  162

 35

 88

 9

 9

 13

  54

 24

 77

  169

 14

 125

 9

 11

 47

  55

 15

 32

  172

 22

 39

 10

 14

 67

  61

 20

 56

  175

 17

 21

 11

 11

 126

  63

 25

 78

  176

 36

 89

 12

 9

 26

  65

 13

 112

  180

 18

 117

 12

 12

 48

  65

 16

 33

  187

 23

 40

 13

 15

 68

  66

 11

 6

  201

 37

 90

 14

 8

 107

  68

 17

 34

  214

 18

 22

 14

 10

 14

  68

 26

 79

  216

 38

 91

 14

 16

 69

  69

 14

 18

  229

 14

 9

 16

 8

 118

  72

 10

 121

  231

 24

 41

 16

 13

 49

  72

 21

 57

  240

 39

 92

 17

 10

 27

  77

 27

 80

  255

 40

 93

 18

 9

 108

  78

 14

 113

  256

 25

 42

 18

 17

 70

  83

 22

 58

  260

 26

 43

 19

 14

 50

  86

 28

 81

  280

 41

 94

 20

 8

 3

  94

 18

 35

  284

 19

 23

 20

 11

 28

  94

 23

 59

  294

 15

 10

 22

 11

 15

  95

 15

 19

  295

 42

 95

 22

 18

 71

  97

 29

 82

  319

 27

 44

 24

 15

 51

  101

 15

 114

  320

 43

 96

 26

 19

 72

  102

 30

 83

  338

 44

 97

 28

 10

 109

  107

 24

 60

  359

 45

 98

 29

 12

 29

  108

 19

 36

  374

 46

 99

 30

 9

 4

  111

 31

 84

  411

 47

 100

 31

 16

 52

  113

 12

 7

  432

 48

 101

 32

 20

 73

  117

 13

 124

  447

 49

 102

 34

 13

 30

  121

 25

 61

  491

 50

 103

 35

 12

 16

  123

 32

 85

  516

 51

 104

 36

 9

 119

  125

 16

 115

  520

 52

 105

 36

 10

 120

  132

 11

 122

  579

 53

 106

  

 

Navigation Table by Quantity of Numbers

  

 

Num-

 # of

 Syst.

 Num-

 # of

 Syst.

 Num-

 # of

 Syst.

bers

 comb.

 #

 bers

 comb.

 #

 bers

 comb.

 #

 6

 5

 1

  13

 117

 124

  22

 172

 39

 7

 3

 11

  13

 157

 8

  23

 49

 76

 7

 5

 24

  14

 10

 67

  23

 94

 59

 7

 9

 2

  14

 19

 50

  23

 187

 40

 8

 5

 12

  14

 43

 31

  24

 54

 77

 8

 8

 25

  14

 69

 18

  24

 107

 60

 8

 14

 107

  14

 78

 113

  24

 231

 41

 8

 16

 118

  14

 169

 125

  25

 63

 78

 8

 20

 3

  14

 229

 9

  25

 121

 61

 9

 5

 45

  15

 13

 68

  25

 256

 42

 9

 9

 13

  15

 24

 51

  26

 68

 79

 9

 12

 26

  15

 55

 32

  26

 137

 62

 9

 18

 108

  15

 95

 19

  26

 260

 43

 9

 30

 4

  15

 101

 114

  27

 77

 80

 9

 36

 119

  15

 294

 10

  27

 151

 63

 10

 7

 46

  16

 14

 69

  27

 319

 44

 10

 14

 14

  16

 31

 52

  28

 86

 81

 10

 17

 27

  16

 65

 33

  29

 97

 82

 10

 28

 109

  16

 125

 115

  30

 102

 83

 10

 36

 120

  16

 132

 20

  31

 111

 84

 10

 51

 5

  17

 18

 70

  32

 123

 85

 10

 72

 121

  17

 37

 53

  33

 133

 86

 11

 5

 64

  17

 68

 34

  34

 136

 87

 11

 9

 47

  17

 136

 116

  35

 162

 88

 11

 11

 126

  17

 175

 21

  36

 176

 89

 11

 20

 28

  18

 22

 71

  37

 201

 90

 11

 22

 15

  18

 43

 54

  38

 216

 91

 11

 36

 110

  18

 94

 35

  39

 240

 92

 11

 66

 6

  18

 180

 117

  40

 255

 93

 11

 132

 122

  18

 214

 22

  41

 280

 94

 12

 6

 65

  19

 26

 72

  42

 295

 95

 12

 12

 48

  19

 52

 55

  43

 320

 96

 12

 29

 29

  19

 108

 36

  44

 338

 97

 12

 35

 16

  19

 284

 23

  45

 359

 98

 12

 48

 111

  20

 32

 73

  46

 374

 99

 12

 113

 7

  20

 61

 56

  47

 411

 100

 12

 132

 123

  20

 133

 37

  48

 432

 101

 13

 8

 66

  21

 37

 74

  49

 447

 102

 13

 16

 49

  21

 72

 57

  50

 491

 103

 13

 34

 30

  21

 151

 38

  51

 516

 104

 13

 48

 17

  22

 40

 75

  52

 520

 105

 13

 65

 112

  22

 83

 58

  53

 579

 106

 

 

PART I

 

Lottery systems with a single guarantee

 

 

System # 26 : Guaranteed 3-win if 3 of the numbers drawn are in your set of 9 numbers Copyright 2020 ©

   

 

Winning possibilities

 

 Guessed

  5

  4

  3

  2

  %  

  5

 1

 -

 5-8

 2-6

 9.52

 

 -

 3

 3-6

 2-6

 25.40

 

 -

 2

 4-7

 2-6

 50.79

 

 -

 1

 7-8

 2-4

 12.70

 

 -

 -

 10

 1

 1.59

  4

 -

 1

 4

 5-6

 7.14

 

 -

 1

 3

 5-6

 19.05

 

 -

 1

 2

 6-8

 19.84

 

 -

 1

 1

 10

 1.59

 

 -

 -

 6

 3-5

 11.11

 

 -

 -

 5

 4-5

 22.22

 

 -

 -

 4

 6-7

 19.05

  3

 -

 -

 3

 0-3

 4.76

 

 -

 -

 2

 4-6

 33.33

 

 -

 -

 1

 6-8

 61.91

  2

 -

 -

 -

 5

 2.78

 

 -

 -

 -

 4

 27.78

 

 -

 -

 -

 3

 69.44

 

 

This system is MATHEMATICALLY MINIMAL with respect to the main guarantee. It also guarantees at least four 3-wins if 4 of your numbers are drawn. If all five drawn numbers are within your selected 9 numbers, then you will either hit a 5-win or up to three 4-wins and additional prizes, or ten 3-wins; the chance of getting at least a 4-win plus a number of smaller prizes is 98.41%. The system is well balanced: Each number is in either 6 or 7 combinations as seen from the table below. It also has an additional nice property: The combinations are as `apart' as possible; in this case, every two combinations differ in at least two numbers, which also provides maximum coverage of quadruples. The complete system would require 126 combinations.

 

  Number(s)

 Occurrences

2, 3, 4, 7, 8, 9

7

1, 5, 6

6

 

 

1.

 1

 2

 3

 7

 8

 

 7.

 2

 3

 4

 5

 8

2.

 1

 2

 4

 7

 9

 

 8.

 2

 3

 5

 7

 9

3.

 1

 2

 5

 6

 7

 

 9.

 2

 3

 6

 8

 9

4.

 1

 3

 4

 5

 6

 

 10.

 2

 4

 6

 7

 8

5.

 1

 3

 4

 8

 9

 

 11.

 3

 4

 6

 7

 9

6.

 1

 5

 6

 8

 9

 

 12.

 4

 5

 7

 8

 9

 

 

... … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

(The full text of this part of the book contains 105 other systems)

 

 

PART II