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Best Yielding Progressive Cover 6/49 Lotto Unique 3's™
by Colin Fairbrother

Perhaps you have wondered like me as to the minimum number of lines or blocks it would take to get 100% Coverage of the Sixes for a Three win but without repeating any Threes and progressively maximizing the Coverage of the Sixes. This is no trivial task and one that some thought lay outside the bounds of reasonable calculation on a modern fast computer. It probably can be done in RAM memory alone using arrays and storing just the results but I chose to progressively write to a database and once you do that things slow down significantly.

Putting it in perspective you can easily generate in about a minute a 3 if 3 for the maximum 496 lines without repeating a Three which is a Cover for the Sixes in a 6/49 game in lexicographic order. Previously, I did the cover in 450 Combs using last number order or progressively increasing the Pool with no repeat Threes but this Cover made no attempt to maximize the coverage of the sixes in the least number of lines. An improvement to 397 lines without repeat Threes was obtained using lexicographic order and with progressive best coverage of the Sixes and this was further improved using some tricks of the trade to 365 as given to the date of this post. 

I am advocating that you will get better results in Lotto by using a design with no repeat combinations of 3 integers (Threes), using all the integers and maximizing the coverage of the Sixes because for each combination added : _

20 more unique Threes, 15 more unique Fours, 6 more unique Fives and 1 more unique Six go to increasing your chances of winning.
.

progressively the set is optimized for best coverage to minimize the number of combinations used in accordance with Cover convention.
.

no gimmicks or distortions are used unlike the majority of Covers or Wheels with Guarantees where anything goes as long as it reduces the number of combinations used with no regard for the decreased yield ie multiple and higher wins are sacrificed to give a lower number of combinations to guarantee a lower prize.
When this Lotto Play Set or 365 Line Cover is analyzed using my own Lotto Number Set Analyzer for all 13,983,816 possibilities more detail is obtained than shown below. eg 63 groups for when a Four is obtained with a varying number of Threes. This is simplified to 21 groups with a range given for the 3's. 
The correct interpretation is important; it is not a history but rather a categorization of the possibilities for 1 draw. If the biggest category or grouping is no result then on the day that is your most likely result; it is an effective way of measuring the spread and magnitude of the prize groupings. This is the method the traditional  "Coverists" use to test their wares so they can't knock it!

An incorrect interpretation (ie treating it as eqivalent to a history of draws) is easily shown to be distorted. Testing the playset against all and only the possible 13,983,816 combinations and treating them as draws gives at 52 draws per year something like 268,919 years. There is  no guarantee that all the possible combinations would be draws in that time frame. In other words over 13983816 draws there will be lots of repeat sixes and if you divided the possibilities into two arbitrary groups of 6,991,908 then for each draw a six is just as likely to be in one group as in the other!

Further, when you do this type of analysis and proportionately decrease the groupings any set gives about what probability suggests which is unrealistic. The table below shows groupings for 1 draw and which group the single relevant number falls in is random but more likely to be in the larger groupings. This correlates with the results borne out by realistic tests against actual Lotto histories that show the more you concentrate your numbers or repeat the subsets and the more you restrict the available integers then the more you reduce your chances of winning in the short term which is the way we all play Lotto! 

From the analysis of my C(49,6,3,6)=365 Lotto Cover with Unique Threes™ you see everything is in accordance with the odds. Your chances of getting a Three in a 6/49 Lotto game are 1 in 56.655927 so for 365 combinations played one would expect an average of 6.44 wins per draw. Looking at the report from my analyzer below we see the dominant grouping at 2,096,805 possible draws or some 15% is that for 7 wins and that is as it should be. The single Three win accounts, as it should do, for only 2346 possible draws or 0.01678% and you can see you are more likely to get 13 right at 7955 draws or .05689%. The results are easily seen as a classic normal or Bell Curve.

Repetition of the subsets in World Record Unique 3's 3 if 6 Cover

The question arises when pursuing the usual purported objective of a Cover ie to eliminate a no win for any draw result in the least number of lines, that in Lotto isn't the overall yield better improved by allowing a few no win results to exist? After all the prize for a Three win is just a token of some $2.00. Wouldn't one prefer an increase in the multiple win groupings rather than satisfy what is really a stupid objective in a game of chance of making a pittance prize certain by distorting the Normal Distribution in increasing the lowest single prize group?

Regards
Colin Fairbrother

ps 1 This is a first attempt and if you are so inclined easy to beat by a line or two. Don't post the enumeration for the plagiarists to pounce on - let me know by PM or email, I can easily verify by a few data specific questions.

Author	Message
Colin F Lotto Systems Tester Creator & Analyst To dream the impossible dream ... Joined: September 30 2004 Location: Australia Online Status: Offline Posts: 678	Topic: Best Yield Progressive Cover 6/49 Lotto Unique 3's Posted: August 08 2008 at 10:36pm
	Best Yielding Progressive Cover 6/49 Lotto Unique 3's™ by Colin Fairbrother Perhaps you have wondered like me as to the minimum number of lines or blocks it would take to get 100% Coverage of the Sixes for a Three win but without repeating any Threes and progressively maximizing the Coverage of the Sixes. This is no trivial task and one that some thought lay outside the bounds of reasonable calculation on a modern fast computer. It probably can be done in RAM memory alone using arrays and storing just the results but I chose to progressively write to a database and once you do that things slow down significantly. Putting it in perspective you can easily generate in about a minute a 3 if 3 for the maximum 496 lines without repeating a Three which is a Cover for the Sixes in a 6/49 game in lexicographic order. Previously, I did the cover in 450 Combs using last number order or progressively increasing the Pool with no repeat Threes but this Cover made no attempt to maximize the coverage of the sixes in the least number of lines. An improvement to 397 lines without repeat Threes was obtained using lexicographic order and with progressive best coverage of the Sixes and this was further improved using some tricks of the trade to 365 as given to the date of this post. I am advocating that you will get better results in Lotto by using a design with no repeat combinations of 3 integers (Threes), using all the integers and maximizing the coverage of the Sixes because for each combination added : _ 20 more unique Threes, 15 more unique Fours, 6 more unique Fives and 1 more unique Six go to increasing your chances of winning. . progressively the set is optimized for best coverage to minimize the number of combinations used in accordance with Cover convention. . no gimmicks or distortions are used unlike the majority of Covers or Wheels with Guarantees where anything goes as long as it reduces the number of combinations used with no regard for the decreased yield ie multiple and higher wins are sacrificed to give a lower number of combinations to guarantee a lower prize. When this Lotto Play Set or 365 Line Cover is analyzed using my own Lotto Number Set Analyzer for all 13,983,816 possibilities more detail is obtained than shown below. eg 63 groups for when a Four is obtained with a varying number of Threes. This is simplified to 21 groups with a range given for the 3's. The correct interpretation is important; it is not a history but rather a categorization of the possibilities for 1 draw. If the biggest category or grouping is no result then on the day that is your most likely result; it is an effective way of measuring the spread and magnitude of the prize groupings. This is the method the traditional "Coverists" use to test their wares so they can't knock it! An incorrect interpretation (ie treating it as eqivalent to a history of draws) is easily shown to be distorted. Testing the playset against all and only the possible 13,983,816 combinations and treating them as draws gives at 52 draws per year something like 268,919 years. There is no guarantee that all the possible combinations would be draws in that time frame. In other words over 13983816 draws there will be lots of repeat sixes and if you divided the possibilities into two arbitrary groups of 6,991,908 then for each draw a six is just as likely to be in one group as in the other! Further, when you do this type of analysis and proportionately decrease the groupings any set gives about what probability suggests which is unrealistic. The table below shows groupings for 1 draw and which group the single relevant number falls in is random but more likely to be in the larger groupings. This correlates with the results borne out by realistic tests against actual Lotto histories that show the more you concentrate your numbers or repeat the subsets and the more you restrict the available integers then the more you reduce your chances of winning in the short term which is the way we all play Lotto! From the analysis of my C(49,6,3,6)=365 Lotto Cover with Unique Threes™ you see everything is in accordance with the odds. Your chances of getting a Three in a 6/49 Lotto game are 1 in 56.655927 so for 365 combinations played one would expect an average of 6.44 wins per draw. Looking at the report from my analyzer below we see the dominant grouping at 2,096,805 possible draws or some 15% is that for 7 wins and that is as it should be. The single Three win accounts, as it should do, for only 2346 possible draws or 0.01678% and you can see you are more likely to get 13 right at 7955 draws or .05689%. The results are easily seen as a classic normal or Bell Curve. Repetition of the subsets in World Record Unique 3's 3 if 6 Cover The question arises when pursuing the usual purported objective of a Cover ie to eliminate a no win for any draw result in the least number of lines, that in Lotto isn't the overall yield better improved by allowing a few no win results to exist? After all the prize for a Three win is just a token of some $2.00. Wouldn't one prefer an increase in the multiple win groupings rather than satisfy what is really a stupid objective in a game of chance of making a pittance prize certain by distorting the Normal Distribution in increasing the lowest single prize group? Regards Colin Fairbrother ps 1 This is a first attempt and if you are so inclined easy to beat by a line or two. Don't post the enumeration for the plagiarists to pounce on - let me know by PM or email, I can easily verify by a few data specific questions.
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Colin F Lotto Systems Tester Creator & Analyst To dream the impossible dream ... Joined: September 30 2004 Location: Australia Online Status: Offline Posts: 678	Posted: January 01 2010 at 10:01pm
	Ion Saliu posted a NON-PROGRESSIVE 412 line Ion Saliu 412 Lines Cover with distinct Threes on RGL on 1 January, 2010 which is 47 lines more than my record. The breakdown on repetition of the Ones and Twos is given below: Colin Fairbrother
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Administrator Admin Group 8888 Joined: September 30 2004 Online Status: Offline Posts: 44	Posted: January 30 2010 at 5:30pm
	On January 30, 2010 Jaera punlished a NON-PROGRESSIVE 371 line Cover on RGL here with Unique Threes.
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Colin F Lotto Systems Tester Creator & Analyst To dream the impossible dream ... Joined: September 30 2004 Location: Australia Online Status: Offline Posts: 678	Posted: February 11 2010 at 6:13pm
	February 12, 2010 GC posted a NON-PROGRESSIVE 358 line Cover in RGL which he passed off to his eternal shame as progressive. A simple check at 68 lines revealed only 11,298,393 combs covered or 80.79621% when 11,716,391 or 83.78536% is easily achieved in for example CoverMaster Colin Fairbrother
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