Colin Fairbrother

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Manually Constructing a Best Lotto Wheel or Cover
by Colin Fairbrother

An understanding of what will deliver the best results on average for a set of numbers played in Lotto can be obtained by manually constructing the set. Let's use the Classic Lotto game with a Pool of 49 integers and a Pick of 6 integers.

Most Lotto games do not pay on getting 1 integer correct in the set you play but if they did and the Pool was 48 with a Pick of 6 then the following 8 lines cannot be improved on ie the coverage of the Ones is 100% and it is not possible to achieve this with a lesser number of lines: -
 
01 02 03 04 05 06
07 08 09 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36
37 38 39 40 41 42
43 44 45 46 47 48
 
It is also isomorphic which means we can randomize the integers and the coverage of the Ones at 100% is the same so it can be used as a template.
 
For a Pool of 49 there are 13,983,816 possible combinations of 6 integers compared to 12,271,512 for a Pool of 48. See Table of various Combinations obtainable for a given Pool. Just one extra integer has increased the number of combinations by 1,712,304, see Table of extra Combinations for an increase in Pool size.
 
To extend our Lotto Design beyond 8 lines there are 1,712,304 possibilities to cover the remaining One (ie 49) but if we restrict our selection to not duplicating a combination of two integers it is considerably less. We see that {1,2}, {1,3}, {1,4}, {1,5} and {1,6} already have an appearance but {1,7} does not. Choosing {1,7} means we now have the integers 1 and 7 appearing twice. Looking at the first available integer greater than that paired with 7 we see that {13,14}, {13,15}, {13,16} {13,17} and {13,18} have an appearance so let's opt for {13,19}. Looking at our set we see that both {1,7} and {13,19} appear in the first column and the next integer is 25 which we can pair with the as yet unused integer 49.
 
The 9th line could then be 01 07 13 19 25 49 but there are plenty of other possibilities that give the same coverage of the Ones. However, all Pick 6, Pool 49 Lotto games do not pay on getting one integer correct so let's go to the first paying subset which is a combination of Three integers and let the One's and Two's fall as they may with no regard for bunkum "balancing" which is the stuff of numerology rather than that of best yield. So far we have 2,312,716 CombSixes covered or 16.53852% coverage with 180 distinct Threes in the 9 lines.

The thought may have occurred to you that rather than repeating 5 integers on one line we spread them over 2 lines. Locking our first 7 lines and initializing the 8th line with something like 10 42 43 44 45 46 and the 9th line with something like 10 45 46 47 48 49 the last 2 lines are easily optimized in CoverMaster. An example to use as a template for randomization is given below with 1 7 42 43 44 46 and 41 45 46 47 48 49 for the last two lines, which now covers an extra 45 CombSixes to give a Coverage of 16.53943%. Yes, I agree the probability of those 45 occurring is about the same as winning first prize!

01 02 03 04 05 06
07 08 09 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36
37 38 39 40 41 42
01 07 42 43 44 46
41 45 46 47 48 49 

For the 10th line let's consider which one will give the best extra coverage of the Sixes for a Three prize without repeating a Three and using a transparent manual method.
 
The best known record coverage for 9 lines is 16.5394% attributed to Adolf Muehl and for 10 lines it is 18.22231%, attributed to Normand Veilleux each being the first known person to achieve this. See Record Partial Coverings.This is not as big a deal as it used to be with the current speed and RAM memory of current computers as for around this number of line sets it is quite easy to evaluate all possibilities. The 10 line record can be easily replicated using John Rawson's CoverMaster program optimizer - (Download here and send request for zip unlock to honest.John@ntlworld.com) - after seeding with say, the 10 lines below using similar to the 9th line, a 1 and 5 integers in the 2nd column which gives 18.22006% coverage: -
 
02 03 04 47 48 49
05 06 07 44 45 46
08 09 10 41 42 43
11 12 13 38 39 40
14 15 16 35 36 37
17 18 19 32 33 34
20 21 22 29 30 31
23 24 25 26 27 28
01 02 05 08 11 14
01 03 06 09 12 15
 
After optimization in CoverMaster we get from the above set the following set with a coverage of 18.22231% of the Sixes: -
 
04 06 07 44 45 49
17 18 19 32 34 46
01 02 05 07 08 14
11 13 33 38 39 40
23 24 25 26 27 28
20 21 22 29 30 31
02 03 10 35 36 48
08 10 16 41 42 43
14 15 16 36 37 47
01 03 09 12 15 43
 
... but is it better in yield than the original 10 lines? Over 100 draws both sets are most likely to give: -

No Wins        82

Three Wins    17

Four Win         1

If you transpose using CoverMaster you can rearrange this set with the same coverage to: -

01 02 03 04 05 06
07 08 09 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
01 31 33 34 40 43
05 32 38 41 42 43
06 31 32 36 37 47
02 34 35 36 38 48
04 39 44 45 46 49

Has replacing the lines - 

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

- really achieved anything as far as Lotto yields are concerned apart from some jumbling of the numbers for an imagined gain?

Is this telling us something?

Can this set be arrived at in a transparent manual way?

Can we use this to work out a way of testing a sample of the possibilities for the next line?

Is there any point to optimization beyond next best play if it doesn't improve the yield in Lotto? See a comparison here?

 Colin Fairbrother



This article first published on the web 28/12/2009 and has been viewed 11654 times.
The author may be contacted at fairbros@bigpond.com. To generate sets of Lotto numbers to play for most Lotto games around the world go to LottoToWin ® where for a US$5.00 annual subscription you can store the numbers as well, to check later for wins. The numbers produced have no duplicate subset combinations of two integers (if applicable) or three integers, use all the integers for the particular Lotto game and maximize the coverage of the potential winning main number. The articles on this site have been selected from the many at the forum LottoPoster
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