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Multiple Win Covers or Wheels
 LottoPoster Forums : NUMBER SETS TO PLAY FROM DIFFERENT METHODS : Multiple Win Covers or Wheels
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Colin F
Lotto Systems Tester Creator & Analyst
Lotto Systems Tester Creator & Analyst
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To dream the impossible dream ...

Joined: September 30 2004
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Quote Colin F Replybullet Topic: Covers or Guarantee Wheels with Multiple Wins
    Posted: October 17 2007 at 10:10pm
 
Covers or Guarantee Wheels in Lotto
with Multiple Wins
by Colin Fairbrother
 
In the Lotto community that is interested in the structure of a set of numbers played the emphasis has always been on producing the minimum set of lines to ensure a single win for a given Lotto game. You don't have to be too long in this area of interest and doing some serious analysis to find that to be distortionary. My emphasis is looking at things from the perspective of Percentage Return or Yield over a reasonable number of draws for the set of numbers played and an honest realistic analysis free
as much as is possible of mathematical terminology. To that end if considering a 6/49 Lotto game with a ticket cost of $0.50 and payout of $2.00 for a single three win we have to consider multiple lowest level prize wins to be ahead in anything more than 3 lines.
 
It has to be said that this field of interest has and still is dominated by the con-artists, pretentious pricks, numerologists and the loony brigade. As I have pointed out before this site won't have a bar of them and on that point my position has hardened for I see them as a bunch of intransigent, obdurate ignoramuses - recalcitrants to the will of reason even when occasionally they can comprehend it. I am encouraged and thankful to receive an email now and then complementing me on maintaining a Lotto Forum where reason and knowledge prevail and a disassociation is made from the lunatic Lotto fringe group and their befuddled nonsense.
 
In the discussion that follows for Lotto games with more than a Pool of 22 integers everything is predicated by first achieving the necessary hits in the pool of 22 integers which are a subset of the integers used in the particular Lotto game. The odds for this happening follow from the general odds for the Lotto game - there is no improvement possible above that of optimizing the threes without duplication in the set of numbers you play, using all the integers and maximizing the coverage. What really happens is that what is achieved naturally by Random Selections is done constructively - such as minimizing repetition of Fives. In the long term the relatively high repetition of Fives in System/Full Wheels works its way out - in the short term it works against you and that's a good enough reason to look for something different. It really comes down to a matter of style, orderliness and simple know how as to how you play your numbers.
 
A good starting point when considering multiple wins is to consider a Pool of 22 for a Pick or Selection of 6. Without any knowledge of Covers you would in any serious analysis of the Ones, Twos, Threes, Fours and Fives that are possible from this Pool of 22 become aware of some remarkable structural details. Most notable is that from the 20 threes that can be formed from a six, 77 lines can be constructed that contain the 1540 distinct threes ie they are packed into the minimum possible set of lines. (77x20=1540). In mathematical circles where they like to attribute increments of knowledge to someone or another this is known as a Steiner set after the Swiss Jakob Steiner around 1853. Usually, in Cover parlance this is written as C(22,6,3,3)=77 but more informative would be C(22,6,3,3,1)=77 where the 1 is not assumed and indicates the minimum prize win.
 
You can easily produce this Cover. An easy check to establish there are no duplicates is to test against the 22c3=1540 possible unique or distinct Threes. In my first attempt at constructing C(49,6,3,6)=365 with no duplicate subset Threes I tried rather than Lexicographic order Last Number order. Naturally, after completing Pool 22 I had the Steiner set of 77 lines; after 450 lines full coverage for 1 Three was obtained without any duplicate CombThrees.

With no computers in the 1850's you can try reproducing this cover. Noting you can't repeat a combination of 3 integers you
could start as follows lexicographically (numerical order) or increasing the Pool progressively (Last Number Order, LNO)

  1  01 02 03 04 05 06      1  01 02 03 04 05 06
  2  01 02 07 08 09 10      2  01 02 07 08 09 10
  3  01 02 11 12 13 14      3  03 04 07 08 11 12
  4  01 02 15 16 17 18      4  05 06 09 10 11 12
      . . .                              . . .
      . . .                              . . .
 77  13 14 16 17 20 22     77  13 14 19 20 21 22
 
Considering a hypothetical but very possible 6/22 Lotto game and the 7,315 possible CombFours then the minimum prize level is 4 CombThree wins. So the same 77 lines Cover could be written as C(22,6,3,4,4)=77. For the possible 26,334 CombFives then the minimum prize level becomes 10 Three wins - so, the same set could be described as C(22,6,3,5,10)=77. For the possible 13,983,816 CombSixes the minimum prize level is 20 CombThree Wins and so the same set could be written as C(22,6,3,6,20)=77. (The highest prize level, if you are very, very lucky, for one of the 77 lines could be the Jackpot)
 
An interesting exercise is to actually see when you would get these 2,464 blocks that give 20 Threes which occur in 3.3% of the 74,613 combinations of six that can be formed from a pool of 22. You can do this easily in Excel, Access, VB or C# by first enumerating the combinations then doing a find. If you constructed your Cover in lexicographic order the first should be at index 502 showing 1 2 3 7 17 20 - if so inclined you can find the 2,463 others. In passing I mention that 74,613 sixes is only around half of 1% of the 13,983,816 sixes that can be formed from a pool of 49.
 
The C(22,6,3,6,20)=77 I find particularly interesting as if we set the cost of playing the 77 lines as $38.50 then for a payout of $2.00 for a Three win you are actually making a profit of $1.50  in the Pool of 22. Obviously it is easier to get 3 right in a Pool of 22 which may be a sub-set of a Pool of 49 but why the emphasis on C(22,6,3,3,1)=77 which produces a loss of $36.50?
 
The Percentage Return (Wins/Cost x 100) from this set of 77 Combs for the multiple wins is better in each case when compared to the best Cover for a single win but only the
C(22,6,3,6,20)=77 gives winnings in excess of the extra cost
.
 
           Cover          Percentage Return      Profit $     Loss $
                                          (if success)  (failed)
 C(22,6,3,4,1)=46     2/23 x 100 =   8.7%    (21.00)     23.00    
 C(22,6,3,4,4)=77   8/38.5 x 100 =  20.8%    (30.50)     38.50
 
 C(22,6,3,5,1)=22     2/11 x 100 =  18.2%     (9.00)     11.00    
C(22,6,3,5,10)=77  20/38.5 x 100 =  51.9%    (18.50)     38.50

 C(22,6,3,6,1)=15    2/7.5 x 100 =  26.7%     (5.50)      7.50
C(22,6,3,6,20)=77  40/38.5 x 100 = 103.9% 
        1.50      38.50
 
(For those who like to have a punt on the Gee Gees the last line is enough for them to ask - "Why is it a certainty?". Your embarrassed answer would be - "Well, it isn't! It's the government you see.")
 
Not all comparisons work out as well as our maximum threes compacted set. eg C(10,6,3,3,1)=10 and C(10,6,3,3,2)=18. On a closer examination taking the cost of a ticket as being 50¢, a Three win as $2.00 and assuming you got 3 from the Pool of 10 right then for the 10 lines you are $3.00 out of pocket compared to $5.00 for the 18 lines! The usual comparison made is that you save $1.00 over playing 2 x 10 lines with the same numbers in the same draw. 
 
Within the context of Covers a set can be produced in just 15 lines that guarantees a Three win every draw for a Pool of 22 integers but with duplicate Threes. The problem is the lowest Pool for an actual Pick 6 Lotto game is 25 and the best Cover with duplicates to guarantee a Three win is 22 lines. Without duplicates I can do the guaranteed Three in 37 lines and because there are no duplicate Threes I am maximizing my multiple wins?
 
The odds for one play in a Pick 6, Pool 22 Lotto game are: -
Three Win 1 in 6.66
Four Win 1 in 41.45
Five Win 1 in 777.22
Six Win 1 in 74,613
So if you played 41 lines you would average a Four win every draw with maybe a few Three wins thrown in.
 
The real question is do you need that guaranteed win every draw for getting three of the integers correct by playing 77 lines when playing 41 lines will give you an average Four win per draw? Generally, isn't it better to have a higher yield or percentage return and have the occasional non winning draw? 
 
Regards
Colin Fairbrother
 
C(22,6,3,6,1)=15
 
a1 a2 a3 a4 a5 a6
1 4 5 6 7 8
1 2 3 5 7 8
1 2 3 4 6 8
2 3 4 5 6 7
9 11 16 17 18 19
9 11 12 17 19 22
9 11 12 16 18 22
9 10 11 13 15 20
9 10 11 13 14 21
10 13 16 17 18 19
10 12 13 18 19 22
10 12 13 16 17 22
12 14 15 20 21 22
14 15 17 18 20 21
14 15 16 19 20 21
 
C(22,6,3,6,1)=24 No Duplicate Subset Triples
 
a1 a2 a3 a4 a5 a6
1 9 11 17 18 19
1 5 8 13 17 21
1 7 19 20 21 22
1 2 9 10 13 16
1 4 8 12 15 16
1 2 3 4 5 6
2 3 8 13 19 20
2 3 10 17 18 22
2 5 7 11 13 14
2 4 7 15 17 21
2 5 12 16 19 22
3 10 12 13 14 21
3 4 16 18 19 21
3 8 11 14 16 22
3 6 7 9 14 18
3 5 9 15 16 20
4 6 7 8 13 22
4 5 9 14 21 22
4 11 15 18 20 22
5 8 10 14 18 20
6 12 17 18 20 21
6 10 11 15 16 19
7 8 9 10 11 12
13 14 15 16 17 18
 
C(22,6,3,3,1)=77
 
a1 a2 a3 a4 a5 a6
1 2 3 4 5 6
1 2 7 8 9 10
3 4 7 8 11 12
5 6 9 10 11 12
5 6 7 8 13 14
3 4 9 10 13 14
1 2 11 12 13 14
3 5 7 9 15 16
4 6 8 10 15 16
1 6 7 11 15 17
2 5 8 12 15 17
2 4 9 11 16 17
1 3 10 12 16 17
2 3 10 11 15 18
1 4 9 12 15 18
1 5 8 11 16 18
2 6 7 12 16 18
3 6 8 9 17 18
4 5 7 10 17 18
13 14 15 16 17 18
2 4 7 13 15 19
1 3 8 14 15 19
1 6 9 13 16 19
2 5 10 14 16 19
3 5 11 13 17 19
4 6 12 14 17 19
8 10 12 13 18 19
7 9 11 14 18 19
1 5 10 13 15 20
2 6 9 14 15 20
2 3 8 13 16 20
1 4 7 14 16 20
7 9 12 13 17 20
8 10 11 14 17 20
4 6 11 13 18 20
3 5 12 14 18 20
4 5 8 9 19 20
3 6 7 10 19 20
11 12 15 16 19 20
1 2 17 18 19 20
8 9 11 13 15 21
7 10 12 14 15 21
4 5 12 13 16 21
3 6 11 14 16 21
2 6 10 13 17 21
1 5 9 14 17 21
1 3 7 13 18 21
2 4 8 14 18 21
1 4 10 11 19 21
2 3 9 12 19 21
7 8 16 17 19 21
5 6 15 18 19 21
2 5 7 11 20 21
1 6 8 12 20 21
3 4 15 17 20 21
9 10 16 18 20 21
3 6 12 13 15 22
4 5 11 14 15 22
7 10 11 13 16 22
8 9 12 14 16 22
1 4 8 13 17 22
2 3 7 14 17 22
2 5 9 13 18 22
1 6 10 14 18 22
2 6 8 11 19 22
1 5 7 12 19 22
9 10 15 17 19 22
3 4 16 18 19 22
1 3 9 11 20 22
2 4 10 12 20 22
5 6 16 17 20 22
7 8 15 18 20 22
4 6 7 9 21 22
3 5 8 10 21 22
1 2 15 16 21 22
11 12 17 18 21 22
13 14 19 20 21 22
 
 
Lotto Draws have no relationship to one another; the integers serve just as identifiers. Any prediction calculation on one history of draws for a same type game is just as irrelevant as another.
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