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# covering theory and covering design theorem

Posted on 2003-12-02
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there is a program called lottogenius at www.lottogenius.com

it works on the covering theory and covering design theorem

has anybody got any code in any programming language  (vb preferred) so that
i can see how this works?
i'd like to make my own lotto genius.

Is there any mathematical background to lottery wheels?
Yes, there is. Lottery wheeling is based on Covering theory, and Covering Design theorem:
Covering Design C(v,k,t,m,l,=b) is a pair (V,B), where V is a set of v elements (called points) and B is a collection of b k-subsets of V (called blocks), such that every m-subset of V intersects at least l members of B in at least t points. It is required that v >= k >= t and m >= t. The case m > k is also a valid case. B can be a multiset.
When applied to lottery:
v   total numbers in the design (amount of numbers you play)
k   amount of numbers drawn
t    min match t   (t of m)
m  min match of m (t of m)
l    lambda (min l times t of m)   (it is always 1and usually omitted)
b   number of blocks   (number of games)
For example,  C(44,6,3,5,154) describes wheel with 44 numbers designed for "pick 6" lottery that guarantees 3 matches if 5 officially drawn balls are within the group of 44 numbers you have played. Finally, the wheel size is 154 games.
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Question by:jxharding
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Thibault St john Cholmondeley-ffeatherstonehaugh the 2nd earned 250 total points
ID: 9857489
I found a forum once that was discussing building lottery wheels. Unfortunately I can’t find it now, it may be blocked from my server – many are including your link.
One solution was reached by defining all the triples (assuming you want to match 3).
The triples were split into two heaps – if a match is found in one heap you don’t need to include a triple from the other.
A program then grabbed pairs of triples and identified any others that were covered by this group. There was mention of a ‘greedy’ algorithm and ‘hill climbing’. With a whole batch of these groups (the wheel), a search was done over the entire set of possible draws, listing the number of hits within each group in the wheel. Those with fewer hits were manually analysed to build them into other groups ie. 2 or 3 groups merged to become a single group and the whole wheel re-tested.
The best wheel I’ve seen for match 3 from 6 from 49 is on here, where there is a competition running.:

http://lottery.merseyworld.com/Wheel/Wheel.html

I’m not sure that it was notes from the creator of this wheel that I was reading.
I’ve run this 163 wheel through a test program and found only two redundant numbers, but I don’t think I can get anywhere near creating a wheel myself.

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Expert Comment

ID: 10153665
Thanks jxharding.

Did you find anything more out?
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Author Comment

ID: 10153693
no, i did get hold of some pdfs with HECTIC algebra, real PHD stuff.
but yours kinda got me on the road.
i dont think i'll ever get it though ,:)

thanks!
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LVL 17

Expert Comment

ID: 10153706
oh well, back to working for a living I suppose :7)

If I do turn anything up I'll try to post it in here.
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