MINIMALITY

Now we take the opportunity to mention one very important quality of all lottery systems in this book. It 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 our systems use the minimum known number of tickets. These systems are combinatorial objects that have been extensively studied, and most of the systems in this book are not likely 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 some of the systems 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 once more at our example, System # 20. It has been proven that the minimum number of tickets is 20 for the given guarantee. In other words, if you want to play 10 numbers and you want a guaranteed 4-win if 4 of your 10 numbers are drawn, then you need to play at least 20 tickets. System # 20 achieves the guarantee in exactly 20 tickets, and this is the minimum possible number of tickets. That is why we call such a system mathematically minimal. The fact that the guarantee cannot be achieved in fewer than 20 tickets means that nobody can ever improve this system.

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