For long term browsers of this forum the gem of text below comes from 2005 when Anastasios had his first and only software running Lotto Architect. The discussion was initiated by Bobijohn a frequent and valuable contributor to the forum. This post got 'lost' when Anastasios transferred from his original forum to this current one. The wheel constructions discussed were those that were capable of being devised within the Lotto Architect system. Nowadays we have of course Wheel Generator to do all this work  I wonder in fact if some of the concepts discussed here may have prompted Anastasios to go and develop his Wheel Generator  for it can quite easily handle number groups at any L setting and of course any desired guarantee level. Covermaster was at that time the standby for wheel makers  Wheel Generator has now superseded this old tool at every level Serotic or matrix constructions as in Wheel Generator being one example.
For your interest.
Bobijohn 07232005
The following analysis is being offered here to show the power of this program and also to show how a User may approach a lotto Game. This also serves to dispel any possible myth that if you can't win with an 80% wheel you sure not going to win with a 50% wheel. I believe they are two different "things" derived completely independently of each other.
All the numbers are presented so anyone can work through the analysis. They are from my Local Pick 5/37 Lotto. The winning numbers for that draw are the real winning numbers.
Let us assume we have performed our analysis (in this case from a HCD system) and now have the following set of N=16 numbers3,6,11,12,13,14,15,16,20,22,23,29,32,35,36 and 37.
Group A (1 thru 9 = 2 balls)  3,6
Group B (10 thru 19 = 6 balls)  11,12,13,14,15,16
Group C (20 thru 29 = 4 balls)  20,22,23,29
Group D (30 thru 37 = 4 balls)  32,35,36,37
From a separate analysis of our history data base files we have determined that at least one of the winning numbers will be in each of the above decade groups approximately 28% of the time. This is true for my local Lotto. Because this is a pick 5 game one of these four decade groups must also contain 2 winning numbers. What this means is that 72% of the time all 5 winning numbers will fall in only three, or less, of the decade groups.
Incidentally, since we do not have Opencover wheels for many large input groups of numbers the above method is one way to break the group down and use smaller wheels for which we do have open cover options.
Based on the above we decide to wheel 3 groups at a time. There is only 4 ways to arrange three groups at a time out of four groups. We also decide to use 4 if 4 wheels. For this example I have chosen one Closecover wheel (L=1) for each number grouping for base comparison and, similarly, two sets of OpenCover wheels (L=0.8 and L=0.5) for each grouping. These would be as follows:
L=1 or 100%  L=0.8 = 80%  L= 0.5 = 50%
Wheel ABC = 12 balls requires 113 tickets  80 tickets  50 tickets
Wheel ACD = 10 balls requires 51 tickets  34 tickets  21 tickets
Wheel ABD = 12 balls requires 113 tickets  80 tickets  50 tickets
Wheel BCD = 14 balls requires 230 tickets  161 tickets  101 tickets
Total tickets (before dupes) 507 tickets  355 tickets  222 tickets.
Dupes removed in Calc 1 19 tickets  10 tickets  8 tickets
Tickets remaining  488 tickets  345 tickets  214 tickets
Note: The comparable Closecover wheel for N=16 balls requires 405 tickets for a guarantee of at least one 4 hit if at least 4 winning numbers are in the input universe as compared to the 488 tickets above. The more even the distribution of balls within each group the closer this percentage difference seems become. Also, the smaller the total number group and individual groups seems to generally lessen the percentage difference.
The actual winning numbers for this lottery were the following> 3,6,14,15 and 34. We got 4 out of 5. That is two winning numbers in group A and two winning numbers in group B. Nothing in group C and nothing in group D.
Now let's look at the wheels we played. No filtering has been performed. With this grouping of numbers we have a guaranteed 4 number win with the Closecover (L=1) wheel ABC and a guaranteed 4 number win with the CloseCover (L=1) wheel ABD. What about the two opencover wheels of L=0.8 and L=0.5?
Here is the summary of hits for all 3 wheels.
 L=1  L=0.8 L=0.5
Match 0  113  79  49
Match 1  224  159  98
Match 2  120  81  48
Match 3  29  26  17
Match 4  2  0  2
Match 5  0  0  0
What happened? The L=1 Wheel had two tickets with 4 hits as expected by the guarantee. The L=1 (80% cover wheel) missed both potential 4 hits!! The L=0.5 (50% cover wheel) collected on both potential 4 hits. I'll leave it to LottoArchitect to discuss what criteria we might use (other than budget) to help decide which opencover wheel to choose and also to provide us with a few words of explanation on these results.
With a little practice with this great program the above analysis can be performed very quickly and easily. I Have found it to be fun and, more importantly, educational. By all means extend this analysis  what about other arrangements of winning numbers amongst the input groups (Tipjust change the winning numbers in the data base and run the compare function) or, what if you had five winning numbers spread out in your input groups? Finally, extend the wheeling to include the other closecovered ratios provided and compare the results. There is much to be learned from these types of analyses.
Just for the record this analysis was performed using LottoArchitect v2.2
I hope this has been helpful.
Good luck everyone.
Bobijohn
relowe 07232005
Bobijohn,
This is a great post and demonstrates the power and capability of Lotto Architect. It also provides some new insight into ways we can play our lottery  I thank you for it.
Your tens grouping study is quite fascinating and suggests to me that one could extend the principle by using other grouping arrangements  say 7's, 8's or 9's depending on the number of balls in the lottery to give an even distribution. The other factor of course is to apply some intelligent filtering where some lines could be eliminated, although as the LA man says, this then destroys the guarantee on the wheel. But overall this program offers one the ability to perform these sorts of studies quite simply, and better still from what I see proposed with LA2.3 it just gets better.
thanks again
relowe
Bobijohn 07242005
Hi relowe
You are most welcome  and thank you for the kind response. Yes, this theme lends itself to many good variations. IMHO it is one of the best strategies for capturing multiple 2nd and 3rd tier prizes assuming you play the appropriate wheel (eg: 3 if 3, 3 if 4 and 4 if 4) and of course have the winning numbers in your input group. In fact, it seems quite good at capturing at least one prize higher than the wheel target in the case of 3 if "x" wheels. This maybe due to the inherent overlap feature  I don't know as I am not a mathematician. I am sure we would all be most interested to see your research results if you feel comfortable publishing them.
Good Luck
Bobijohn
lottoarchitect 07242005
And again another sound analysis Bobijohn! I think you can create a strategy guide with all those ideas!
Well, I'll comment on some points you mention above.
QUOTE
This also serves to dispel any possible myth that if you can't win with an 80% wheel you sure not going to win with a 50% wheel. I believe they are two different "things" derived completely independently of each other.
This is absolutely true. Building a minimal wheel L=0.8, even if it is based on a L=0.5 does not mean it will contain even partially most of the tickets in L=0.5. In fact, it is very unlikely to even have the same tickets among them. For example, have a look at the following opencover wheels
C(14,6,3,3,0.52)=10

1 2 3 9 11 12
1 3 4 10 13 14
1 5 7 10 11 13
1 6 7 8 12 14
2 3 6 7 8 13
2 4 5 7 9 14
2 4 5 8 10 12
3 5 6 9 10 14
4 5 6 11 12 13
5 8 9 11 13 14
C(14,6,3,3,0.81)=17

1 2 3 5 12 14
1 2 3 9 11 13
1 2 4 7 10 14
1 4 5 6 8 13
1 5 7 9 10 11
1 6 7 8 10 12
1 6 8 9 11 14
2 3 4 6 7 11
2 4 7 8 9 12
2 5 6 9 10 14
2 5 8 11 13 14
2 6 10 11 12 13
3 4 5 9 11 12
3 4 8 10 11 14
3 5 7 8 12 13
3 6 7 9 13 14
4 9 10 12 13 14
If you compare them, they do not contain the same tickets. This is a general rule. It is possible to make these two wheels have very few tickets in common (1 ticket the same can always exist) and still retaining the same coverage without affecting the properties of the wheel. This can be done by rearranging the tickets so at least one ticket in L=0.81 can match a ticket in L=0.52. In short the build process is totally independent for every wheel. About the loss of 4# hits in the L=0.8 wheel, there are a few things that can be done like the above solution and also to convert all wheels in frontcovered ones. You can check what is this at Covermaster (Reports> V Summary). This is a way to ensure winnings in all opencover wheels, given your winning numbers fall in the range 1X. This solution can avoid the unlucky event of the L=0.8 you have experienced but on the other hand it tends to reduce the spreadability of the hits; provides more hits if winning numbers fall in the initial pointers 1X and fewer beyond pointer X, thus in a sense you give priority to your selected numbers.
Besides that, your analysis is very cool! I enjoyed reading this and is a good approach.
cheers
lottoarchitect
Hyperdimensional 07242005
Hi Bobijohn,
I agree with your analysis, the decade filter offers a new variety of strategies good enough to obtain secondary prizes,
I'll complement the analysis with a comparison with the Pick 5 16 number wheel 4if4.
If you create the wheel in Lottoarchitect and check the results of the winning numbers then you obtain 1 of 4 and 24 of 3 with 405 tickets.
The advantage of the wheel is 488405=83 tickets less, the guarante of 4 is 100%, the disadvantage of wheel is that 2 prizes of 4 is only 10.49% and 3 of 4 is 0.38%, it can't win more than 3 of 4.
Regards
Bobijohn 07242005
Hi LottoArchitect
Thank you for the kind words and good followup explanation of the L=0.8 case.
In the example, I reference the number of duplicates encountered in each case when using all four wheels and it is a small percentage of the total ticket sets3.7%, 2.8% and 3.6%.
Your reference to front covered wheels is most interesting. Just something else to follow up on !! LOL
Regards
Bobijohn
Bobijohn 07242005
Hi Hyperdimensional
Thanks for the response and adding the N=16 4 if 4 case. I should have done that parallel to the others instead of just mentioning it. To summarize for other readers, the extra 83 tickets (L=1 case) gives you the benefit of a guarantee of two 4 hits instead of one plus (most likely) additional two and three hits.
Did you use CoverMaster to derive the 10.49% and 0.38% figures?
Thanks again for your input
Regards
Bobijohn
Hyperdimensional 07252005
QUOTE
Did you use CoverMaster to derive the 10.49% and 0.38% figures?
Hi Bobijohn,
Yes, I used Covermaster to check the wheel Pick 5, (5,16,4,4)=405 guarantees.
Regards
Bobijohn 07252005
Hi Hyperdimensional
Thanks. I guess now we just need a way to do this for the combined wheels. Oh Well!!
Regards
Bobijohn
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