What is a Lotto Wheel?

This is one of the finest explanations of Wheeling you can find.  It is by Normand Veilleux.

Original Article Date:   06/25/1995

   ... and just as useful today.

With new wheeling information included for:

... 04/1998
... 04/2000
... 11/2000
... 06/2004.

Let me show you how wheeling can be helpful.  Instead of choosing only 6 numbers, you decide to choose 7 "lucky" numbers.  It is possible to make exactly 7 different tickets with those 7 numbers.  Most people I know buy all 7 tickets in such a case.  When the tickets are purchased in a Lotto 6/49, this gives them exactly 432,824 chances out of the 13,983,816 possible tickets of winning something.  This represents 3.10%.

Now, the question that the first wheel inventor may have asked was probably something like:  "Can I buy less than the 7 tickets and still have a 3.10% chance of winning?"  It turns out that you can indeed.  Actually, in this particular case you can select any 4 of the 7 tickets and you will still win 3.10% of the time!  So, why waste the extra 3 dollars?  This is called a 3 on 3 system since you are guaranteed of matching at least 3 numbers on at least one ticket even if only 3 of the draw numbers are in your 7 selected numbers.

Of course, 3 dollars is not much, but lets have a look at the case where you choose 12 numbers instead of 7.  In this case, there are a total of 924 different tickets that can be made with 12 numbers.  If you were to buy all 924 tickets you would win 2,069,298 times out of 13,983,816 in a Lotto 6/49 (that's 14.80%).  If you were to use a wheeling system for 12 numbers that was a 3 on 3, you would still win in 2,069,298 cases, but it would only cost you 15 dollars instead of 924!

Here is a set of 15 tickets that contains 12 numbers and that is a 3 on 3 system:

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

All you have to do to use such a system is replace the numbers 1 to 12 by your selected numbers.  For example if you liked the number 22, you could replace 1 by 22 or 10 by 22.  It doesn't matter which one you replace, so long as you replace each number only once.  There is no catch twenty two!  :-)  Changing the numbers around like this preserves the minimum win guarantee.  Mathematicians say that the resulting set is isomorphic to the original set.  Their structure is the same and hence their probabilities of winning are also identical.

If instead of using such a wheel the lottery player decided to buy 15 random tickets from the 924 possible tickets that can be made from the 12 selected numbers, he or she would end up with the same cost, but most probably no minimum guarantee.  There is only a small fraction of all sets of 15 tickets with only 12 numbers in them that have a minimum win guarantee.  Why leave it to randomness to choose your numbers when you can use the knowledge of more than 100 years of wheeling?

Mathematicians use a different term for wheeling.  They call it coverings.  There are many known minimal coverings, but in certain cases the minimal covers are not known and there is a lot of research being done to find them.  Recently, someone asked in sci.math what was the minimum amount of tickets needed to be guaranteed that one of your tickets would match at least a 3 on 6 in a Lotto 6/49.  Someone had discovered a set of 308 tickets that could do that.  I mentioned that I had found a set of 207 tickets that could do it.  The discussion that resulted led us to find a set of 174 tickets that has the same minimum win guarantee.  We are not sure if this set is minimal, though.

Here are recent developments in this wheel.

... As of 04/1998 -- three years after Normand's article was written, the wheel was reduced to 168 tickets.

... In 04/2000 -- after two more years, it was reduced to 165 tickets.

... In 11/2000, it has been further reduced to 163 tickets.  This is the current world record for this wheel.  The new wheel is designed by Rade Belic.

... In 06/2004, the wheel still stands at 163 tickets.

This indicates how difficult it is to construct efficient wheels and improve on them.  It takes time and dedication.

...CDEX

Anyone can choose 12 "lucky" numbers, but not everyone can select tickets with those 12 numbers in order to maximize their probability of winning at the lowest possible cost.  When you use a wheel, you basically take advantage of the analysis that someone else has already done.  You save time and money.  For example, now that you know that using the 3 on 3 wheel with 12 numbers gives you the same probability of winning than buying all 924 possible tickets, I hope you'll never buy more than 15 tickets if you ever want to play with only 12 numbers.

The idea here is not to spend more money because you are using a wheel.  But instead, to use a wheel to make efficient use of your Lotto budget.  The wheels tell you the maximum you should spend in a given circumstance to achieve a certain goal; and if you exceed that amount, then you are basically throwing your money out the window.  But isn't that what lotteries are all about anyway!

The reason I think wheeling is so popular is that people like to be in control.  With a wheel, you choose the numbers, you make the decisions, you are in control.  With randomly selected tickets you are merely an observer, just like buying any other pre-printed tickets.

Normand Veilleux