Combinations Played in Pick 6, Pool 49 Lotto

by Colin Fairbrother

Rarely does a layman get the opportunity to show members of academia with high mathematical qualifications to be utterly wrong. Such is the case for me in exposing the utter fallacy of contriving an increased coverage for a lower prize in Lotto to achieve a Cover or Guaranteed Prize for the set played but an overall lower Yield.

While proving the irrelevance of using a particular Lotto game history to construct numbers to play has given me some satisfaction it can rightfully be regarded as overkill because of the Independence of Events rule and certainly is not as sweet as that obtained from proving highly intelligent and qualified people to have been utterly wrong in advocating Lotto players should play various Covers or Wheels with Guarantees.

From the time I became aware of Covers or Wheels around 2000 there was something about them that came across as dubious and contrived - how right I was. For a Pick 6, Pool 49 Lotto game with ticket costs at 50¢ or £1 and the lowest prize at around $4 or £10 in respectively the USA and the UK an outlay of $81.50 or £163 was recommended for a 163 line Cover. Given the odds of getting three integers in a line correct are 1 in 57 for this game how could anyone justify spending more than $28.50 or £57 per draw if the objective was to have at least an average of one win per draw. The answer apparently was in the desire for the guaranteed pittance no matter what the cost.

In the natural order of numbers played in a wagering scenario with multiple prize levels like Lotto one would expect: -

- As the number of lines played is increased then so should one's chances of both getting a higher prize and multiples of the lower prizes increase and indeed a point be reached where getting just one of the lowest prize ceased to be a possibility.

. - Duplicating a paying combination of the integers in Lotto for the set played reduces your chances of getting one of those prizes.

. - For a Lotto game from which a random selection of five or six integers is made then the best opportunity for success is to play at least a set of numbers which includes all the integers. For a Pool 49 Pick 6 game this is a minimum of 9 lines and for Pick 5 it is a minimum 10 lines.

Consider a Pick 6, Pool 49 Lotto game. There are -

49c6 or

49!/(43! x 6!) or

the inverse of 6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44 or

13,983,816

49!/(43! x 6!) or

the inverse of 6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44 or

13,983,816

chances of getting first prize. Similarly, the possible combinations of three integers are the inverse of 3/49 x 2/48 x 1/47 or 18,424. If you played one line such as 01 02 03 04 05 06 then for the first prize you have covered only one line of the possible 13,983,816 combinations of six integers but for the combinations of three integers you have 20 in your one line which covers 260,624 of the 13,983,816 possibilities meaning each contains at least one of those 20 CombThrees.

From the previous paragraph after say, 01 02 03 04 05 06 the next line with the next best coverage or play of 260224 could be 07 08 09 10 11 12 but there are many more to choose from as long as an integer is not in your first line.

There is a convention for describing Covers which if adhered to makes it easy to find any Cover from among maybe hundreds of files.

C(12,6,4,5,3)=22 means for a Pool of 12 and Pick of 6 in this hypothetical Lotto game if you get 6 correct, at a minimum 13 prizes will be obtained and if 5 correct at a minimum 11 prizes. The construction of this set is done such that the 220 combinations of Three integers from a Pool of 12 are repeated twice in the 22 line set meaning all the 924 Combinations of 6 integers from a Pool of 12 have at least two matching CombThrees.

A special case referred to as a **Steiner** is where the Cover or minimum number of lines multiplied by the Prize Combination in a line is the same as all the possible Prize Combinations for the Pool. A famous Steiner is C(22,6,3,3,1)=77 where the maximum combinations of three integers from a Pool of 22 ie 1540 is the same as the 77 lines multiplied by the 20 combinations of three integers in a Pick 6 line.

Consider a hypothetical Pick 6, Pool 12 Lotto game in which you have set your objective to play 22 lines. The possible unique combinations of three integers from a Pool of 12 is 220 so for 22 combinations of six integers you need to repeat the CombThrees twice for all to have the same multiple. You could try a time consuming method of manually setting it out making sure you don't repeat a Three more than twice and a Four more than once. (If your Fours are unique then so are your Fives and Sixes.)

A much quicker method for producing this set is to first generate 22 lines next best play 5if6 accepting the first in lexicographic order to give: -

1 01 02 03 04 05 06

2 01 02 03 07 08 09

3 01 02 03 10 11 12

4 01 04 05 07 08 10

5 01 04 05 09 11 12

6 01 06 07 08 11 12

7 02 04 06 07 09 10

8 02 05 06 08 09 11

9 03 04 06 08 09 12

10 03 05 06 07 10 11

11 01 05 06 09 10 12

12 02 03 04 05 07 12

13 02 03 04 08 10 11

14 07 08 09 10 11 12

15 01 03 04 07 09 11

16 02 05 06 08 10 12

17 01 02 04 08 09 12

18 01 02 05 07 10 11

19 01 03 05 08 09 10

20 01 04 06 08 10 11

21 02 03 06 09 11 12

22 01 03 04 07 10 12

2 01 02 03 07 08 09

3 01 02 03 10 11 12

4 01 04 05 07 08 10

5 01 04 05 09 11 12

6 01 06 07 08 11 12

7 02 04 06 07 09 10

8 02 05 06 08 09 11

9 03 04 06 08 09 12

10 03 05 06 07 10 11

11 01 05 06 09 10 12

12 02 03 04 05 07 12

13 02 03 04 08 10 11

14 07 08 09 10 11 12

15 01 03 04 07 09 11

16 02 05 06 08 10 12

17 01 02 04 08 09 12

18 01 02 05 07 10 11

19 01 03 05 08 09 10

20 01 04 06 08 10 11

21 02 03 06 09 11 12

22 01 03 04 07 10 12

If this set set is now optimized by changing an integer up or down to a limit set (in this case 5) and testing for a better coverage the following Cover is obtained C(12,6,4,5,3)=22 with the integers changed in red.

1 01 02 06 04 05 11

2 01 02 03 07 08 06

3 01 02 03 10 11 12

4 06 04 05 07 08 09

5 04 05 09 10 11 12

6 01 05 07 08 11 12

7 02 04 06 03 09 10

8 02 05 03 08 09 11

9 03 04 06 08 11 12

10 03 05 06 07 10 11

11 01 05 03 09 06 12

12 02 03 04 05 07 12

13 02 07 04 08 10 11

14 03 07 08 09 10 12

15 01 03 04 07 09 11

16 02 05 06 08 10 12

17 01 02 04 08 09 12

18 01 02 05 07 09 11

19 01 03 05 08 04 10

20 01 09 06 08 10 11

21 02 07 06 09 11 12

22 01 06 04 07 10 12

For the amount of money you care to wager over say 100 draws your return for the lower prizes can be reasonably calculated according to probability rules. A marked deviation usually indicates some distortion has been introduced and generally, with few exceptions this applies to most so called wheels or covers that have been touted as beneficial in Lotto since the 1980's where the lowest number of lines has been the main priority.

Optimization or maximization as illustrated above is OK. However, when it is used to artificially skew a design such as favoring lower over higher prizes to produce a lower number of lines with an overall lower Yield that is bordering on fraud.

Recreational mathematics goes back to the 1600's so unless you're over 400 years old I doubt that you can lay claim to such a simple design. Let us not forget that while people did not have computers they still had the same or better intelligence and perseverance. Slide Rules were first built in England in 1632 and were still being used by NASA engineers in the space program that put a man on the moon in the 1960's. Lo Shu Magic Squares date from around 2800 BC

The following graph highlights the distortion of optimized sets for a 6/49 Lotto game from around 126 combs onwards especially for the 3'sx1, 3'sx2, 3'sx3 and 3'sx4.

A picture's worth a thousand words. Anyone not able to recognize there is something radically wrong with the above graph compared to the natural curves of the graph below has rocks in their head.

Emphasis must be made that in the chart the sets have used the full Pool of 49 integers - using a partial pool as advocated by a nutty professor can lead to abysmally poor results. A realistic perspective is made by showing the full range for the Percentage Yield indicating that the variation is within a narrow band of some 4 percentage points. Random Selections are below the others up to around the 90 combs mark when the Optimized dips below and again more severely at around the 145 combs and mildly below at the 163 mark. **The 3if6 and 4if6 are noticeably superior to the Optimized set staying together to around the 100 combs mark from where the 4if6 gives superior results up to around the 112 mark**.

- The lowest actual Jackpot Lotto game has a Pick of 6 and Pool of 25 numbers. Leaving out just one of these integers removes a significant 24% of the possible Jackpot winning numbers from consideration.
**Leaving out just one integer removes 12.24% from consideration in a Pick 6, Pool 49 Lotto game - however, if you reduce the pool to 10 integers a whopping 99.998% of the possible winning numbers have been removed from consideration. How Ilyya Bluskov, a Professor of Combinatorics, could promote such a system to Lotto players defies comprehension.**. - Since the 1980's and the publication of a list of Cover tables or Wheels by Ivan Dimitrov subsequently popularized by Gail Howard the guarantee of a prize in Lotto if a template is used for a Pool of numbers less than that of the full pool has been promoted. The effect of reducing the Pool which makes the guarantee non-applicable to the game played is glossed over by a glib statement that says if you get so and so correct from the drawn number in the set promoted you will get such and such a prize.
**An honest approach which I use is to calculate the prize table for the same set using the actual Pool for the Lotto game you are playing.****Then we see a dramatic difference - your chances of getting any prize using a Pool of 10 and 20 lines is only 9% compared to double that of 18% when the full Pool of 49 is used.**

. - If you play just a few integers of the Pool and repeat a paying subset then your prize may be higher when you do get it but it will not be as frequent.
**Real Lotto players playing a reasonable number of lines, say 18 in a 6/45 game and 10 in a 6/49 want the frequency of wins to be as high as possible for the money spent.**If you are a regular weekly player then some years playing a small partial Pool you may do quite well but other years get only a a third or less of what is expected.**Playing the same set each draw and restricting the Pool to say 12 integers and repeating the subset Threes twice which can be easily done manually in 22 combinations of six but keeping the Fours unique is a tested exception to the rule on repetitions regarding yield but you still have a longer Maximum Draws with no wins (as many as 47 draws - ie over 1022 plays - compared with less than half that - 21 draws ie 453 plays - when using the full pool with Unique 3's - in a 6/49 game).**The advantage is lost with a higher partial Pool and repeating subsets.**.** **Lotto is a game of chance not certainty.**A normal Lotto player acquaints himself with the odds usually given on the back of an entry form for say the**lowest prize and any prize**and realizes if the latter is 1 in 54 plays as for a 6/49 game then if playing a random selection 18 lines per draw a win every 3 draws on average can be expected.**The notion that a player wants a certain lowest prize win every draw at nearly 3 times the cost is absurd, nonsensical and something dreamed up by misguided boffins not Lotto players.**If with some tweaking one can get an extra few percentage points in Yield by using all the integers, not repeating paying subsets and maximizing the yield through coverage without distorting the prize distribution then well and good and that is what I recommend.

.- A normal Lotto player is concerned mainly about the size of the First Prize, the odds for getting that prize and how fair the operator is in paying a proportionate prize for the lower prizes relative to the odds. This is best summarized by Yield which is the overall percentage payout for say 100 actual draws repeated say 10 times with different draws or a randomized set. A prize table can also be calculated fairly rounding out the likelihood of each prize group.
**The notion that an increased coverage for a given number of lines or the making of a Cover for the lowest prize in the least number of lines by distorting the prize distribution has been shown to produce an inferior Yield.**

Colin Fairbrother

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Best possible Set of Numbers to play in

Pick 6, Pool 49 Lotto analyzed by grouping of

Win results for all possible 13,983,816 winning numbers

Playing just 28 numbers in a 6/49 Lotto game this Lotto Design gives you nearly a 46% chance of getting a prize each draw! The other designs shown with excessive repeat subsets don't even come close, even when a few extra plays are allowed for the Covers or Wheels with Guarantees. **Those that are within 10% proximity are definitely not Covers or Wheels with Guarantees and effectively this proves wrong that Covers or Wheels with Guarantees are superior for this number of combinations as advocated by the likes of Professor Iliya Bluskov and Gail Howard. Moreover, due to the distortionary nature of maximized Covers, with excessive repetitions, as related to Lotto where the minimum pool is 25, my Unique 3's ™ using the full Pool and maximizing Coverage against the main numbers are always superior by a long margin. **

How does the 54% no win per draw stack up against probability formula? One must be wary here because 13,983,816 possibilities is not the same as that many random draws. Considering 28 plays over 13,983,816 draws is unrealistic and it is unnecessary as over just say, 2,000 draws your chances of getting any prize per draw will be around 52% which also means your chances of getting nothing per draw are 48%. There is a correlation with the difference in the figures being explained through nearly 9% of the possible draws giving multiple wins which when averaged out increases the Win percentage. It's a good measurement of getting something or nothing or the grouping most likely for a draw to occur in and realistically nothing, a Three or a Four is what you are more likely to get.

**Unique 3's™ (and therefore no duplicate Fours, Fives or Sixes)**.**With all integers used in the Lotto game**.**And Coverage of all possible winning sixes maximized**

**"You can lead a horse to water but you can't make it drink!"**

Interestingly, those who have looked through the test results forum would have recognized a familiarity with a post I made back in December, 2004 on a 50% Cover for a 6/49 Lotto game using all integers twice and with the best coverage. At that stage I had not worked out the fatal flaw with optimized Covers or Wheels with Guarantees ie sacrificing your chances of multiple wins by stupidly guaranteeing one pitiful win. Looky here!

Regards

Colin

Colin

ps 1 Having established the principles for the best Lotto Play Sets my objective in starting this web site has been achieved. I hoped with the forum format that there would be some constructive input from others but that has not been the case so I will in due course construct a different portal with a greater emphasis on presenting the information, rather than looking for input from others. I have done a complete makeover of LottoToWin to enable you to play and store these number sets so you can check for wins later and this will be uploaded to commence with the USA PowerBall configuration change in January 2009 from 55 to 59 main numbers and from 42 to 39 for the PowerBall. Practically all Lotto games around the world will be covered and the annual subscription will be a nominal $5.00.

ps 2 Here is a summary of each Number Set with 1 representing the worst: -

- One Six combination is repeated 28 times; 98.13825% of Sixes uncovered.

. - Full System Eight or all the combinations of 6 integers from a Pool of 8; 95.30397% of Sixes Uncovered. How the Lottery Operators can promote the second worst style of play as increasing your chances of winning defies comprehension.

. - Cover or Wheel with Guarantee of a Five Win if 5 of the winning integers from a pool of 9 are obtained; despite allowing an extra 2 combinations to bring it up to 30 still 93.32568% of Sixes are uncovered. The abjectness and absurdity of this Cover is further revealed when it is realized that just 12 of the combinations maximized for coverage give the same uncovered percentage! Far from increasing the chances of getting a Five with a repeat of 54 of the Fives it decreases your chances not to mention the 624 Fours and 916 Threes repeated. The section of the mathematical community that espoused this Cover as being beneficial in Lotto and still continue to do so, should bow their heads in shame. The ignominy for the mathematical community at large is that exposure of this fraud was not done by one of their own but by yours truly, a competent database programmer guy that went to the trouble of checking things out.

. - One Five combination is repeated 28 times; 89.44119% of Sixes uncovered but still better than 5if5in9 Cover.

. - Cover or Wheel with Guarantee of a Four win if 4 of the winning integers from a pool of 11 are obtained; despite allowing an extra 4 combinations to bring it up to 32 still 88.25823% of Sixes are uncovered.

. - One Four combination is repeated 28 times; 83.99215% of Sixes uncovered but still better than a System play or both the 5if5in9 or 4if4in11 Covers.

. - C(14,6,4,5)=30

. - One Three combination is repeated 28 times; 73.48612% of Sixes uncovered but still better than a System play or the 5if5in9, 4if4in11 or 3if3in15 Covers.

. - Cover or Wheel with Guarantee of a Three win if 3 of the winning integers from a pool of 15 are obtained; despite allowing an extra 3 combinations to bring it up to 31 still 74.28749% of Sixes are uncovered. One Three combination is repeated 28 times; 73.48612% of Sixes uncovered but still better than a System play or the 5if5in9, 4if4in11 or 3if3in15 Covers.

. - C(24,6,3,5)=30

. - Cover or Wheel with Guarantee of a Three win if 4 of the winning integers from a pool of 19 are obtained; despite allowing an extra 2 combinations to bring it up to 30 still 69.60506% of Sixes are uncovered. For 28 of the combinations maximized for coverage of the Sixes the uncovered percentage rises to a whopping 91.15284%. Lordy, Lordy how can brain matter be so under utilized?

. - Cover or Wheel with Guarantee of a Three win if 6 of the winning integers from a pool of 28 are obtained; despite allowing an extra 3 combinations to bring it up to 31 still 65.485% of Sixes are uncovered.

. - Partial Cover to 28 combinations of 6 integers generated to give maximum coverage for 3if3in49 but only using pool of 48 to give 65.93677% of Sixes uncovered.

. - Good old Ramdom Selections - 2nd best and putting to shame all the nonsense that has been promoted since the 1980's with 60.33555% uncovered for the 28 combinations.

. - The best - 28 combinations with Unique Threes using the full pool of 49 and with coverage maximized to give only 54.25824% of Sixes uncovered.

in a Pick 6, Pool 49 Lotto game -

an Analysis of Win Groupings for 13,983,816 possible draw results

by Colin Fairbrother

Unlike the 14 other sets of numbers analysed in this series Random Selections vary a bit in their structure. To be seen to be fair in choosing a representative example I have included a table below showing the first 28 draws for the year in 4 USA 6/49 State games. As you can see the variation is small and even the lowest percentage covering of any prize at 40% comfortably beats the best of the Covers or Wheels with Guarantees by 7% and the worst by 35%. I have chosen Wisconsin 2007 as being the closest to the average to do the analysis on.

The figures given here are from my own Lotto set analyzer. There are 37 Prize Groupings for this set; the table below reduces this to 14 by giving the main prize grouping and a range for the lesser prizes. eg Of the 368,211 combinations with a single Four win 228,246 have no other prizes.

Repeats Analysis Random Selections sample:

For the Ones we see that only 46 of the 49 are used and the idea of "balancing" or having them appear the same number of times is proven absurd. For the Threes which do pay unlike the ones 10 are repeated twice and 1 Four is repeated. So, to avoid the repetitions which reduce your chances for a prize use my Best Unique 3's ™.

Regards

Colin Fairbrother

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in a Pick 6, Pool 49 Lotto game without Maximizing Coverage -

an Analysis of Win Groupings for 13,983,816 possible draw results

by Colin Fairbrother

The reason I include this playset is to show that simply constructing your number set from a lexicograpic or numerical order enumeration and making sure you don't duplicate a CombThree and you do include all the Pool integers you end up with a set that makes a mockery of all the Covers with a reduced Pool at around 28 lines. The construction is simple - just enumerate lexicographically the combinations of six integers and you will find you have all 49 integers in 29 lines. Delete the 28th line and that's it - you have a set with 34% coverage. The old saying, "Couldn't see the wood for the trees" comes to mind.

ID | LexID | Comb | Comment |
---|---|---|---|

1 | 1 | 01 02 03 04 05 06 | |

2 | 54956 | 01 02 07 08 09 10 | |

3 | 96115 | 01 02 11 12 13 14 | |

4 | 126006 | 01 02 15 16 17 18 | |

5 | 146901 | 01 02 19 20 21 22 | |

6 | 160816 | 01 02 23 24 25 26 | |

7 | 169511 | 01 02 27 28 29 30 | |

8 | 174490 | 01 02 31 32 33 34 | |

9 | 177001 | 01 02 35 36 37 38 | |

10 | 178036 | 01 02 39 40 41 42 | |

11 | 178331 | 01 02 43 44 45 46 | |

12 | 220593 | 01 03 07 11 15 19 | Already used {1,3,4} {1,3,5} {1,3,6} {8,9,10} {11,12,13) |

13 | 231953 | 01 03 08 12 16 20 | Already used {8,9,10} {11,12,13} {12,13,14} |

14 | 242496 | 01 03 09 13 17 21 | |

15 | 252262 | 01 03 10 14 18 22 | |

16 | 324893 | 01 03 23 27 31 35 | |

17 | 327421 | 01 03 24 28 32 36 | |

18 | 329652 | 01 03 25 29 33 37 | |

19 | 331610 | 01 03 26 30 34 38 | |

20 | 341318 | 01 03 39 43 47 48 | |

21 | 370322 | 01 04 07 12 17 22 | |

22 | 380346 | 01 04 08 11 18 21 | |

23 | 392020 | 01 04 09 14 15 20 | |

24 | 400596 | 01 04 10 13 16 19 | |

25 | 474134 | 01 04 23 28 33 38 | |

26 | 476238 | 01 04 24 27 34 37 | |

27 | 478784 | 01 04 25 30 31 36 | |

28 | 480400 | 01 04 26 29 32 35 | Not used |

29 | 490408 | 01 04 40 44 47 49 |

There are 28 Prize Groupings for this set; the table below reduces this to 13 by giving the main prize grouping and a range for the lesser prizes. eg Of the 362,172 combinations with a single Four win 154,999 have no other prizes.

Regards

Colin

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Colin

or 31 Line Cover C(28,6,3,6)=31

in a Pick 6, Pool 49 Lotto Game -

an Analysis of Win Groupings for 13,983,816 Draw Possibilities

Using 28 integers from a Pool of 49 a noticeable improvement is shown over other Covers but with only 34% Covered after allowing an extra 3 combinations this is still well short of the 46% Coverage for my best set. Looking at the repeat table below we see that 46 Threes are repeated twice and 20 trice and 7 Fours are repeated twice wasting respectively 86 and 7 chances.

Regards

Colin

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Colin

or 30 Line Cover C(19,6,3,4)=30

in a Pick 6, Pool 49 Lotto Game

an Analysis of Win Groupings for 13,983,816 Possibilities

Would you knowingly play a set of numbers in a Pick 6, Pool 49 Lotto game where for no good reason 3 of the combinations of 4 integers are repeated twice and repectively 57 of the combinations of 3 integers are repeated twice and 2 repeated 3 times. Well, that is the case with this Lotto Wheel or Cover which guarantees a Three win if 4 of the integers are correct in a pool of 19 you have chosen out of the possible 49. The decoy with these nonsense Covers, that have been touted since the 1980's by various con-artists and still are, is the guarantee, without consideration of how your chances of winning any prize or prizes are reduced!

Regards

Colin

Colin

Playing 30 Line Cover C(24,6,3,5)=30

or Wheel with Guarantee, 3 if 5 in 24

There are no prizes for 71% of the possible draws compared to only 54% for my best set? Looking at the repeats table we see 14 of the Fours are squandered by repeating twice and the Threes are abominable with 80 repeated twice and 40 repeated 3 times.

Playing 31 Line Cover C(15,6,3,3)=31

or Wheel with Guarantee, 3 if 3 in 15

With 74% of the possible winning combinations giving zilch compared to only 54% for my best set why would anyone play this? Looking at the repeats table we see 14 of the Fours are squandered by repetition and the Threes are abominable with 98 repeated twice, 3 trice, 19 quadrupled and 1 repeated 5 times.

Regards

Colin

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Colin

Analysis of Potential Win Groupings

by Colin Fairbrother

Despite repeating a subset Three 28 times and giving away 3 combinations this set still does better than the Cover or Wheel with Guarantee which gives a Four win if you have 5 integers correct in a Pool of 14. No one in their right mind would play this set; how can the "experts" promote this pitiful Cover as being of benefit in playing Lotto?

There are 53 Prize Groupings for this set; the table below reduces this to 15 by giving the main prize grouping and giving a range for the lesser prizes. eg Of the 204,588 prize groups with a single Four win 9,078 have no other prizes.

Regards

Colin

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Colin

Playing 30 Line Cover C(14,6,4,5)=30

or Wheel with Guarantee, 4 if 5 in 14

This Cover or Wheel with Guarantee is listed as Number 35 by Professor Iliya Bluscov and Number 507 by Gail Howard, albeit slightly better in their way of ranking these things. You see the coverist way of looking at it is that it doesn't matter how distorted you make the set with regard to yield as long as the number of lines, blocks or combinations is less, then it is better. The fact that from all the possible 13,983,816 winning combinations only 21% will give a prize compared to 46% with my best Unique 3's ™ using the full pool and with coverage maximized is simply not acknowledged. **Truly one can describe Covers or Wheels like this to be the best of the worst and their advocates for use in Lotto to be none other than con-artists.**