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Algorithms to Calculate the Lexicographic Order or Combination Sequence Number Specifically for Powerball, Mega Millions, Euromillions

By Ion Saliu, Programmer of Lexicographic Order Software

Learn algorithms, formula, formulas to calculate order, lexicographic, lexicographical.

Written on July 4, 2002 (2 WE).

• There has always been a great interest in combination. There is a branch of mathematics named combinatorics. The lottery and many other gambling businesses rely on combinatorics or the mathematics of combinations. Theory of probability itself wouldn't have been possible without the science of combinations.

Two major difficulties in combinatorics are related to two factors:

1) Determining theoretically the rank (or lexicographical order, or index) of a given combination;
2) Constructing theoretically the combination of a given rank (index).

The first attempt to address the major issues came in 1974. B.P. Buckles and M. Lybanon created a computer program to construct a combination when the rank is known (algorithm ACM #515). It is a great achievement. The authors of that algorithm, however, were unable to reverse the procedure and determine the rank for a given combination.
I was the first to solve the problem. I did it in 2001. I wrote the computer program SEQUENCE. I presented the core of my algorithm in the message:

My algorithm did not cover an important combinatorics field: Powerball/Mega Millions. It is a different and difficult lotto game. It draws 5 or 6 numbers like in any regular lotto game. Then it draws separately a so-called power ball or mega ball that can be equal to or less than any regular number previously drawn.

The two major issues were so difficult that nobody came even close to solving them. I took a superficial look at the issues. I thought I would be able to solve the puzzle quite easily. I moved the task to the back burner because I've always had projects I consider more important than others. Then I forgot about it…

The Powerball lexicographic algorithm didn't work out the first day. If I can't come out with a solution the first day, chances are there is no solution. Almighty Number grants us one unit of Light and one unit of Darkness to carry out important tasks.

You can download the program from the most astonishing software download site in history (see the Resources section or the footer of this webpage).

I verified the combinatorial algorithm with many cases. I believe the procedure is 100% accurate. But, if you think you discovered a miscalculation, please let me know. Please document the error(s) as detailed and as clearly as possible. I believe I can fix the errors – if any. Also importantly, I always express my gratitude to people who discover genuine errors in my software, systems, strategies, theories; e.g. I offer freebies not available to every visitor to my Web site.

My lexicographic software is now at its peak of comprehensiveness and accuracy: LexicographicSets.

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Study algorithm, formula for combination index, lexicographic, lexicographical order by Ion Saliu.

This is a comprehensive list of my writings on the topic of lexicographical order or indexing, including algorithms and software.

Of course, everybody loves to feast on great software, especially when it is free! My combinatorics software (and other categories) is absolutely free to run, for an unlimited period of time. However, only the registered members have a right to download the software. Membership requires a nominal fee — the most reasonable there is to connect to the greatest and most useful software ever created. No kidding! Read the conditions to becoming a registered member: Download Great Free Software: Paid Membership Required.

Algorithm, formula, math, to calculate combination sequence number, order, lexicographic, lexicographical.

First algorithm software to calculate combination lexicographical order in Powerball, Mega Millions.

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