Now, I realize and have pointed out before that for 13,983,816 draws there is no guarantee all the possibilities will be represented and it is certain there will be duplicates. However, an interesting exercise is to compare the results over say 100 simulated draws where the prizes are proportional to their representation. The following table gives the results for four 163 line Lotto Designs: -

 

 

Below is a comparison between the Cover C(49,6,3,6)=163 and the first 163 lines taken from my 365 line Cover with all integers included, no duplicate Threes and Coverage obtained by simply selecting for each line the next best play which does better than the 163 line Cover by 4%. The 365 line cover at any number of lines gives good coverage and may give a better yield than an optimized full or partial cover with duplicate Threes and other than the full pool for the Lotto game. 

 

Colin Fairbrother 

 

How does it fare when tested against Random Selections and the first 207 lines of my Progressive 365 Line Cover with Unique Threes. Pretty much as expected - the optimum density helps out but it is more erratic with a greater Standard Deviation. In the 5 runs of 52 draws it has done better than Random Selections this time but it can't always be relied on to do that. Against my 307 lines it has done noticeably worse which is to be expected as the absence of one integer makes 1,712,304 of the 13,983,816 of possible winning numbers as not possible to contribute a prize.

 

 

For a 6/49 Lotto game one can easily generate a 322 line cover in CoverMaster with repeat Threes. This can be reduced to 319 by running the optimizer after deleting 3 lines and to 118 lines by deleting one line and optimizing. I believe anyone who uses Covermaster with some expertise, even if only to check their own work, would have had, like me, these coverings in their files for many years.

 

A thread was started in RGL on January 2, 2010 to see what lower limits could be achieved for a so called "straight" cover ie one without any obvious indication that merged covers had been used, in the least number of lines. The author. a rather strange chap by the name of Gary Dobner who naively believes he is free to libel good people on public newsgroups with impunity, while hiding under some silly pseudonym such as a or gARY and plagiarising others work, gave as his first contribution simply the first optimization from Covermaster without mentioning that source or the programmer, John Rawson. For these covers where duplicates are allowed, when the lines are sorted by dependence (Combinations Uncovered if removed) and viewed in the dependency chart for a 3 if 6 setting in Covermaster there should be a reasonably smooth curve from the highest dependency to the lowest without any extended plateaus which would indicate an undisguised merge cover.

 

For a Normal Distribution cover as the number of lines is increased so there should be a progressive, reasonably smooth increase in the coverage for the measurement of progress towards at least one Three and after less than 56 lines should be in decline with two Threes taking over. So, if you like a cascading, decaying wave formation which I have done, but not here. This is not a requirement for the covers produced with the least number of lines and no evidence of merging.

 

In the case of the Stojiljkovic & Belic C(49,6,3,6,1)=163 Cover the dominant grouping is the Threes x 1 at 25.71% which is a distortion with the Threes x 2 at 23.32% not greater than the Threes x 1 and the Threes x 3 at a paltry 7.97%. See Chart example for Pool 45  here. The occurrence of the integers when comparing the two merged sets is close but noticeably different with a range from 18 to 21 and 22 on the 21 mark because of the C(22,6,3,3,1)=77.

 

 

 

 

 

 

The last entry at 174 lines which just scraped in on the parameters specified is purely to illustrate that the pursuit of the lowest number of lines by inflating the 3's x 1 is utter folly. The 3's x 2 is just greater at 3,287,977 than the 3's x 1 at 3,259,795. To ram home the point the 3's  x 1 at 3,259,795 has comparable figures in my 365 line Unique 3's™ Cover at about the 14 and 100 combs mark with a peak of 6,484,296 at 53 combs.       

 

 

Colin Fairbrother

 

 

 

Colin

 

 

Colin Fairbrother

 

I think this site is starting new.

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Try it, there are thousands of wheels for free!

 

Best wishes

WeEf