Tutorial - Constructing Serotic coverings X if Y in WG 1.6

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lottoarchitect
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Tutorial - Constructing Serotic coverings X if Y in WG 1.6

Post by lottoarchitect » Sat Dec 20, 2008 11:28 pm

Among the various constructions possible in Wheel Generator, we can construct the famous Serotic coverings offering x if y! In the past, we could easily construct Serotic coverings for Jackpot in Lotto Architect (for more info see viewtopic.php?f=7&t=28), however it was not possible to offer other guarantees, such as 5if6 or 4if6. Now, with the introduction of number groups and the guide mode, this has become possible.

Background
In Serotic coverings, we primarily define some groups of numbers, where we expect to have exactly k numbers correct in each such group compared to the actual draw. For example, a typical Serotic covering with 15 numbers would be:
Group 1 numbers : 01 02 03 04 05 <- expecting to have 2 numbers correct in this group.
Group 2 numbers : 06 07 08 09 10 <- expecting to have 2 numbers correct in this group.
Group 3 numbers : 11 12 13 14 15 <- expecting to have 2 numbers correct in this group.

We can of course assign any numbers we want instead of the pointer numbers. It is important however to note that the groups cannot overlap to their numbers in order to generate a Serotic covering. Also, it is not necessary to request or have the same amount of numbers in each group, but the sum of the requested numbers in all groups should equal the block size played; that means 6 for a lotto 6/49 game. In this example, we have 2+2+2 = 6 so this will be a valid Serotic construction.

Setting up WG for Serotic construction
The standard Serotic Jackpot covering requires 1000 blocks to play (can't be reduced). If we have exactly 2 numbers correct in each group, then we are guaranteed the winning combination among those 1000 blocks. However, this is still quite a lot to play, so it would be sensible to consider ways of reducing this. For example, how would we guarantee a 5 if 6, given the above groups? The process is straightforward in WG 1.6; all we have to do is to setup those 3 groups and set them in "guide" mode. Their range should be set to 2-2. The covering parameters should be v = the total numbers we play with (15 in our example), k = the block size (6 in this example), t = X we request, m = k (the block size, 6 as well). Since Serotic coverings are filtered coverings (means filters drive the possible blocks to be used), we also have to say WG that we primarily are interested in improving the filtered coverage. To do so, switch to the coverages panel and enable the check box "Filtered over overall coverage".

A 5if6 Serotic covering in 82 blocks guaranteeing 5if6, as produced by WG 1.6, when we have the groups' conditions satisfied.

01 02 06 07 13 15
01 02 06 08 14 15
01 02 06 10 11 15
01 02 07 08 12 15
01 02 07 09 11 14
01 02 08 09 11 13
01 02 08 10 12 13
01 02 09 10 12 14
01 03 06 07 13 15
01 03 06 08 11 14
01 03 06 09 12 15
01 03 07 10 11 12
01 03 07 10 13 14
01 03 08 09 14 15
01 03 08 10 12 13
01 03 09 10 11 15
01 04 06 07 14 15
01 04 06 09 11 13
01 04 06 10 12 14
01 04 06 10 14 15
01 04 07 08 11 15
01 04 07 09 12 13
01 04 08 09 13 14
01 04 08 10 12 15
01 05 06 07 11 12
01 05 06 08 11 12
01 05 06 09 13 14
01 05 07 08 12 14
01 05 07 09 12 15
01 05 07 10 11 13
01 05 08 10 13 15
01 05 09 10 11 14
02 03 06 07 12 14
02 03 06 09 11 12
02 03 06 10 13 14
02 03 07 08 11 13
02 03 07 09 13 15
02 03 08 09 12 15
02 03 08 10 11 15
02 03 09 10 11 14
02 04 06 07 11 12
02 04 06 08 13 15
02 04 06 09 12 14
02 04 07 08 13 14
02 04 07 09 11 15
02 04 07 10 12 13
02 04 08 10 11 14
02 04 09 10 13 15
02 05 06 08 11 14
02 05 06 09 12 13
02 05 06 10 11 13
02 05 06 10 12 15
02 05 07 08 11 12
02 05 07 10 14 15
02 05 08 09 11 15
02 05 08 09 13 14
03 04 06 08 12 13
03 04 06 08 13 15
03 04 06 10 11 13
03 04 07 08 14 15
03 04 07 09 11 14
03 04 07 10 12 15
03 04 08 09 11 12
03 04 09 10 13 14
03 05 06 07 11 15
03 05 06 08 11 14
03 05 06 09 14 15
03 05 06 10 12 13
03 05 07 08 13 15
03 05 07 09 11 13
03 05 07 09 12 14
03 05 08 10 12 14
03 05 09 10 12 13
04 05 06 07 13 14
04 05 06 08 12 15
04 05 06 09 11 15
04 05 07 10 11 14
04 05 07 10 13 15
04 05 08 09 12 13
04 05 08 09 14 15
04 05 08 10 11 13
04 05 09 10 11 12

The hits produced by this particular covering (when the groups' conditions are satisfied)

Code: Select all

Details information on 6 hits (Filtered)
---------------------------------------------------------------------------------------------------
                 6        5        4        3        2        1        0    Total        %    Sum %
---------------------------------------------------------------------------------------------------
                 1    0 - 2   3 - 11  26 - 40  19 - 32   5 - 20    1 - 5       82    8,20%    8,20%
                 0        4    4 - 6  25 - 31  27 - 37   6 - 13    2 - 4        7    0,70%    8,90%
                 0        3   5 - 11  19 - 32  23 - 44   3 - 17    1 - 5       90    9,00%   17,90%
                 0        2   6 - 13  17 - 34  22 - 40   6 - 19    0 - 5      337   33,70%   51,60%
                 0        1   8 - 15  17 - 35  17 - 38   7 - 21    0 - 4      484   48,40%  100,00%
       0H      918        0        0        0        0        0        0     1000                  
       0%   91,80%    0,00%    0,00%    0,00%    0,00%    0,00%    0,00%                                                                    
However, if we miss one number from one of the groups, i.e. hit only one, we still get at least a 4 hit! No need to say that we effectively reduced the total blocks to play by 918 (or a 91.8% reduction in cost)!

Expanding the idea of Serotic coverings, what would we get if we request a 4 if 6 for the same groups?
WG 1.6 produced the following Serotic covering in 15 blocks which offers 4if6, when all the groups are fully satisfied.

01 02 08 10 13 14
01 02 09 10 11 15
01 03 06 09 11 14
01 03 06 09 12 13
01 04 07 08 11 12
01 05 06 07 14 15
02 03 06 07 11 14
02 04 06 08 12 15
02 04 08 09 13 14
02 05 07 09 12 13
03 04 07 10 13 15
03 05 08 09 11 15
03 05 08 10 12 14
04 05 06 10 11 13
04 05 09 10 12 14

Its relevant detailed hits analysis:

Code: Select all

Details information on 6 hits (Filtered)
---------------------------------------------------------------------------------------------------
                 6        5        4        3        2        1        0    Total        %    Sum %
---------------------------------------------------------------------------------------------------
                 1        0    0 - 2    1 - 3   6 - 13    0 - 3        0       15    1,50%    1,50%
                 0        2    0 - 1    2 - 5    4 - 7    2 - 4        0       10    1,00%    2,50%
                 0        1    0 - 3    2 - 8    2 - 9    1 - 5    0 - 1      250   25,00%   27,50%
                 0        0        5        1        5        4        0        2    0,20%   27,70%
                 0        0        4    1 - 4    3 - 8    0 - 4    0 - 1       36    3,60%   31,30%
                 0        0        3    1 - 6    1 - 9    0 - 6    0 - 2      197   19,70%   51,00%
                 0        0        2    3 - 8    1 - 8    0 - 5    0 - 2      345   34,50%   85,50%
                 0        0        1    5 - 9    1 - 7    0 - 5    0 - 2      145   14,50%  100,00%
       0H      985      725        0        0        0        0        0     1000                  
       0%   98,50%   72,50%    0,00%    0,00%    0,00%    0,00%    0,00%                           
What we can see here is that on the average case, we'll have a 5 hit 27.50% of the time (the sum % of the 3 top lines) or 2 4-hits 34.50% of the time. Definitely not bad at all when we play only 15 blocks! This always assumes that our groups are completely satisfied by the next draw to come (i.e. having exactly two correct numbers in each group).
In the case we completely fail to the conditions of our groups, the covering is still optimized for the overall coverage offered. In this example, the Serotic covering offers 96.16% 4if6, for as long as we have 6 correct numbers within our selection of 15. As a reminder, the unconditional 15,6,4,6 = 19 blocks and by using this filtered construction, we optimize and gain improved hits in the case the next draw completely satisfies our filters. However, since we cannot usually ensure that our groups will be always fulfilled, WG is aware of that fact and tries to eliminate any loss as much as possible by increasing the overall coverage - a feature that doesn't exist in any other similar software to my best knowledge. This is just one of the unique features of WG that accompanying its powerful engine. :)

And of course, for the first time we are able to construct Serotic coverings that offer multiple prizes (L)!. If we request from WG to produce the above Serotic 4if6 to provide at least 2 4-hits (setting L=2), then we get the following covering in 24 blocks (6 less than doubling the relevant 4if6 in 15 blocks):

01 02 07 08 14 15
01 02 08 09 11 15
01 03 06 09 11 12
01 03 06 10 13 14
01 03 07 08 12 14
01 04 06 10 13 15
01 04 07 09 12 13
01 05 07 10 11 12
01 05 08 09 12 13
02 03 06 09 11 13
02 03 08 10 12 15
02 04 06 08 12 13
02 04 06 10 12 14
02 04 07 08 11 14
02 05 07 10 11 13
02 05 09 10 14 15
03 04 07 09 13 15
03 04 08 10 11 15
03 05 06 07 12 15
03 05 06 08 11 13
03 05 07 09 13 14
04 05 06 07 14 15
04 05 08 09 11 14
04 05 09 10 12 14

with the following detailed hits analysis:

Code: Select all

Details information on 6 hits (Filtered)
---------------------------------------------------------------------------------------------------
                 6        5        4        3        2        1        0    Total        %    Sum %
---------------------------------------------------------------------------------------------------
                 1        0    1 - 3   3 - 12   4 - 15    2 - 6        0       24    2,40%    2,40%
                 0        2    0 - 2    3 - 9   6 - 14    1 - 7    0 - 2       34    3,40%    5,80%
                 0        1    1 - 5   3 - 12   4 - 15    0 - 7    0 - 2      364   36,40%   42,20%
                 0        0        6    4 - 6    4 - 9    2 - 7    0 - 2        7    0,70%   42,90%
                 0        0        5    3 - 8   5 - 12    1 - 7    0 - 3       67    6,70%   49,60%
                 0        0        4   4 - 10   4 - 13    0 - 7    0 - 2      180   18,00%   67,60%
                 0        0        3   6 - 12   2 - 13    0 - 8    0 - 3      217   21,70%   89,30%
                 0        0        2   8 - 13   2 - 10    0 - 7    0 - 3      107   10,70%  100,00%
       0H      976      578        0        0        0        0        0     1000                  
       0%   97,60%   57,80%    0,00%    0,00%    0,00%    0,00%    0,00%                                            
Even more amazing is the coverage offered for L=2 when we completely miss the conditions of the groups! WG loyal to the promise it wants to deliver, also optimized the overall coverage to 91.66% to get 2 4-hits, or 98.60% to produce 1 4-hit, regardless if the actual draw passes our groups' constraints, for as long as we have 6 numbers correct within our 15 selected!

Now you have a deeper insight in the workings of Serotic coverings, its your turn to put WG in fire ;)
cheers
lottoarchitect

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