Algorithm, software to calculate combination lexicographical order for Powerball

Algorithm, algorithms, formula, formulas, formulae, math, mathematics to calculate combination order, index, lexicographic, lexicographical.

The algorithms to calculate the lexicographic order or combination sequence number specifically for Powerball and Mega Millions

By Ion Saliu, Rank-And-File At-Large

The algorithms to calculate the lexicographic order or combination sequence number specifically for Powerball and Mega Millions lotto games. Also, generate the combination from the rank or order for Powerball and Mega Millions.

Written by Ion Saliu 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) of a given combination;
2) constructing theoretically the combination of a given rank.
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 (1 WE). I wrote the computer program SEQUENCE.EXE. 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' 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 algorithm didn't work out in 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.EXE
with the description: “COmbination Lexicographical ORDER”, version 1.00, July 2002 (2 WE).
The program is freeware, of course. Download it from the most astonishing FTP download site in history:
www.saliu.com/infodown.html

I verified the algorithm with many cases. I believe the procedure is 100% accurate. But, if you think you discovered a miscalculation, please let me know. Document the error as detailed and as clearly as possible. I believe I can fix the errors – if any.
Best of luck!
Ion Saliu

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.EXE, 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.EXE or DrawIndex.EXE. 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 the free software from the download site:
LexicographicSets.EXE, the universal permutations, arrangements and combinations lexicographic indexing (ranking);
DrawIndex.EXE - highly automated lexicographical indexing for lotto draw files.
NthIndex.EXE ~ lexicographic indexing superseded by LexicographicSets.EXE;
SEQUENCE.EXE ~ lexicographic indexing superseded by LexicographicSets.EXE;
COLORDER.EXE - lexicographical indexing for lotto and Powerball only;
PermuteCombine.exe, the universal permutations, arrangements and combinations generator for any numbers and words;
Combinations.exe, the universal combinations generator for any lotto, Keno, Powerball game: N numbers taken M at a time, in K steps;
WRITER.EXE, random generator of letters to words, passwords, sentences;

Subject:

Comments:
: : Combination Lexicographical Order, Rank, Index.T This is the comprehensive and lightning-fast algorithm. There has always been a great interest in 'combination sequence number' or lexicographical order. 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. I wrote the computer program SEQUENCE.EXE. I presented the core of my algorithm in the message: 348.html "Combination sequence number" or "Lexicographical Order" Fast Algorithm. My algorithm did not cover an important combinatorics field: PowerBall. 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' 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. The PowerBall algorithm didn't work out in 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. The mathematics of combinations: combinatorics. Description: "Combination Lexicographical ORDER", version 1.00, lexicographic, lex. Don't ask me to give you the source code! I ain't no googoo; nor am I googol. There are so many bastards out there. They live only to steal and be asskicked by JAQK Fowuru. The masochists enjoy that, I dunno why. I want first to offer my algorithm to reputable repositories. I offered my previous algorithm to netlib.org. I don't know what was with them. I don't know how they handled my source code. Are there bastards everywhere, in any imaginable place? How can be some so stupid in their quest to satisfy their sick egos? So, the source code can be had only from a reputable repository. Until then, use COLORDER.EXE. I verified the algorithm with many cases. I believe the procedure is 100% accurate. But, if you think you discovered a miscalculation, please let me know. Document the error as detailed and as clearly as possible. I believe I can fix the errors – if any. The correct formula should take into consideration differences in lotto games formats. One fundamental element of a lotto game format is its largest number. Thus, in a lotto 49/6 game, the CSN for combination 6-7-16-20-28-47 is 6,991,908 (exactly the mid-point in the combination set). The same combination 6-7-16-20-28-47 has an index (CSN) of 45,148,858 in a lotto 69/6 game (PA lottery). Your formulae do not lead to these results, the only correct results. Although not a general-purpose formula, the algorithm I presented a few days ago represents the easiest way to calculate the indexes of any lotto combinations. The algorithm is for the BASIC language, but it can be converted to any programming language. I compiled NTHINDEX.EXE with PowerBasic, especially because of increased speed of execution. strange as it apparently seems to some I too use the ID in my lotto programs. Combinatorial mathematics: The very first rule they learn is this: "In combinations of N elements taken M at a time, M must be less than or equal to N. For it is absurd to try combining 2 elements taken them 3 at a time, or 0 elements 6 at a time." If one has only 2 elements, how can one combine them 3 at a time? One particular value that M can take is 0. "The number of combinations of any N elements taken 0 at a time is 1." But "0 elements taken N at a time is an absurdity." I presented earlier this month of July 2001 an algorithm to calculate the index, or rank, or lexicographical order, or combination sequence number (CSN). The algorithm was 100% precise. It had one problem, however. The speed of execution left much to be desired for monster-odds games, such as Keno (80/20, 70/20, even 70/10). The speed was acceptable for many other lotto games. Even a big game such as the 55/5/42 PowerBall required less than half a minute to determine the CSN. Nevertheless, running the algorithm for Keno games took an eternity! I came back to the original "B.P. Buckles and M. Lybanon" algorithm (algorithm "ACM #515"). I was able to add to it the reverse task: calculate the CSN for any lotto game, but the PowerBall. Later, the PowerBall will be added to the procedure. I recompiled NTHINDEX.EXE to SEQUENCE.EXE, the program that uses the new algorithm. Title: "Algorithms, software to calculate combination lexicographical order for Powerball, Mega Millions". The algorithms to calculate the lexicographic order or combination sequence number specifically for Powerball and Mega Millions lotto games. Also, generate the combination from the rank or order.

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