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

This is to announce the discovery of the algorithms to calculate the lexicographic order or combination sequence number (CSN) specifically for Powerball, Mega Millions, and Euromillions lotto games. Vice versa, generate the combination from the rank or order or index for Powerball, Mega Millions, Euromillions.

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:

• Combination sequence number or Lexicographical Order Fast Algorithm.

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.

It was minutes after the end of the Darkness period. I saw the invisible PowerBall algorithm in a late dream. I tried to implement it several times, to no avail. It was a hot summer morning. Every line of code was melting down soon after its inception. The ice cubes did nothing but add fuel to the fire. I waited for another unit of Darkness. I wrote the algorithm invisibly, in a dream. When I woke up I found a compiled program already. It was named:
COLORDER
with the description: COmbination Lexicographical ORDER, version 1.0, July 2002 (2 WE).

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.

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

• Lexicographic, lexicographical order, index, rank of permutations, exponential sets, combinations.
  This is the most intuitive, if not the best, introduction to the apparently difficult concept of lexicographic ordering (or indexing).

• Lexicographical Order: Lotto, Powerball, Mega Millions, Euromillions.
  This is the most comprehensive and intuitive presentation of the concept of lexicographic ordering (or indexing), including the superior software to tackle the task.

• Combination sequence number or Lexicographic Order Fast Algorithm.
  This is how it all started. I responded to challenges in various newsgroups. I came up with the complementary algorithm to the original algorithm by B.P. Buckles and M. Lybanon" (algorithm "ACM #515"). I offered my lexicographic algorithm bona fide to ACM. A bunch of suckers there tried to make my algorithm look like their creation, not mine!

• Calculate the combination lexicographic order of lotto data files.
  The introduction of DrawIndex, the software that calculates the lexicographic indexes for every combination (draw) in a lotto results file.

  The following are older and somehow outdated materials I wrote on lexicographical order subjects. Most of the older software is outdated. Nothing (in the whole world) can beat LexicographicSets or DrawIndex. Sentimentalism is good only to a close extent.
  

• Index, rank lexicographic order lotto combination.
• Software combination algorithm formula index ordering.
• Combination sequence number or Lexicographic Order Debates.

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.

• Download Science, Mathematics, Statistics, Combinatorial Software, plus much more:
  LexicographicSets, the universal permutations, arrangements and combinations lexicographic indexing (ranking);
  DrawIndex - highly automated lexicographical indexing for lotto draw files.
  NthIndex ~ lexicographic indexing superseded by LexicographicSets;
  SEQUENCE ~ lexicographic indexing superseded by LexicographicSets;
  COLORDER - lexicographical indexing for lotto and Powerball only;
  PermuteCombine, the universal permutations, arrangements and combinations generator for any numbers and words;
  Combinations, the universal combinations generator for any lotto, Keno, Powerball game: N numbers taken M at a time, in K steps;
  WRITER, random generator of letters to words, passwords, sentences;
  

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