Over many years of working with wheels (covering designs) I've
come up with some theories that may prove useful to some
members of the lottery community. None of this is to say,
"I'm right and you're wrong." and if you're sure I'm wrong you
might not fully appreciate what I'm trying to communicate.
Contained within every six number combination are found the
following prize winning opportunities . . .
20 ways to make a 3 number winning ticket
15 ways to make a 4 number winning ticket
06 ways to make a 5 number winning ticket
06 ways to make 5 plus a bonus number into a winning ticket
01 ways to make a 6 number winning ticket
This is powerful enough, that any two tickets with all twelve
numbers different form a 3if5in12number2combination wheel and
of course cover 3if6in12n2c as well. Three tickets with all
eighteen numbers different cover better than 80 percent of the
combinations necessary to guarantee a 3if6in18n prize if all
six of the winning numbers are to be found among the 18 being
As wheels grow larger to capture better prizes and/or to bring
more numbers into play, combinations form that contain some of
the same sub-combinations as other combinations in the wheel.
Here are the pointer numbers for a simple 4if5in9n3c wheel...
1-2-3-4-5-6 4-5-6-7-8-9 1-2-3-7-8-9 notice how the three
sub-combinations 1-2-3, 4-5-6, 7-8-9 are all used twice in
this balanced wheel. If you were to have the three winning
numbers 3-6-7 you could not win, that's why this is called a
4if5 wheel. On the other hand if you had any combination of 5
of the 9 numbers on the wheel you would be 100% guaranteed one
4 number winning ticket and if you had all 6 of the winning
numbers among the 9 not only would you have a guaranteed 4
number win, you would also have about 65% coverage of what it
takes to guarantee a 5# win. Not bad for three tickets.
Anyway the point was the three redundant sub-combinations in
this little wheel. As wheels grow these tend to multiply to
the point where we're buying a lot of redundant coverage for
little prizes we're not all that interested in winning.
I used to have a lottery program called LottoMan by Scott Piel
one of the first to actually make wheels on the fly from
scratch. On an old computer you could actually watch the
percentage of coverage grow as combinations were tested. What
caught my eye, was how at first the combinations covered a big
percentage of what was necessary to cover, at 80-90% coverage
a wheel had far fewer combination then the wheels I was using
from books and lotto software.
I mean, I was seeing 90% coverage of 4if6in18n in only 24
combinations when 42 are required for 100%. Unfortunately the
program could not be stopped at that ideal point and continued
the build by throwing random combinations at the problem which
would result in a bloated wheel of about 60 combinations.
Still I realized even back then, 90% coverage at 2/3rds the
cost to play was the direction I wanted to go.
[side note] Yes, anytime you play fewer tickets your overall
odds of winning are lower when computed by the simple process
of dividing your number of tickets into the total number of
combinations. I personally believe you are best off buying
the most efficient coverage your budget allows to get the most
bang for your buck.
Remember the 4if5in9n2c wheel above, well here it is at the
front of a 4if6in12n6c wheel. . .
The larger wheel has been front loaded with a smaller tighter
wheel at no additional cost (abet possible a fraction of a few
points less efficient in some prize tiers) yet with much
greater potential for those who rank their numbers from most
likely to least and put them on the wheel in that order.
In the same sense, I found within the area of savings by
playing wheels of 80-90% coverage rather then the bloated 100%
wheels, there was room to add a combination here and there to
turn clusters of combinations into stand alone quality wheels
within the larger wheel.
Now sure, you might say any collection of combinations you
find within a wheel will test out to percentages of coverage
and be called an open-cover wheel, just as above I showed you
how any two or three tickets form a 3if wheel. The difference
is the quality and efficiency of coverage of the wheels within
wheels. Just as I found you can put the 4if5in9n3c wheel into
the 4if6in12n6c wheel, you can put two of those 4if6in12n6c
wheels into a 3if6in24n wheel and still achieve 95% of
3if6in24numbers coverage in only 15 combinations!
Many wheels these days are made by putting a 3if3 wheel next
to a 3if4 wheel to make an overall 3if6 cover because if all
six winning numbers fall among those being played, at least 3
must fall onto the 3if3 or if not then 4 must fall onto the 3if4
half of the wheel. This is wonderful strategy if you want to
win a 3# prize. If you want to do better, then all the
numbers must fall onto one side or the other. Even though it
is always slim odds to find all the winning numbers on a small
wheel, you have a better chance of it happening when they are
of equal size and strength, rather then a small 3if3 standing
alone from a 3if4 with no numbers in common to both sides.
Despite the obvious value of open cover wheeling (playing a
high percentage of coverage by eliminating some redundancy at
a reduced cost to play) some experts insist even the smallest
hole in a cover is a sure bet to lose the prize. I always
have to wonder why they play if they are that unlucky. On the
other hand, it is also possible to make a wheel that operates
on sort of an either/or principle. Where you have say a 5%
chance of a 5# win or an 85% chance of winning a 4# prize if
all six winning numbers are among those being played and if
the 10% chance of a loss kicks in you win 10 or more 3# prizes
as a consolation instead, all for a 3rd less then playing a
100% 4 if wheel. Not to mention all the times you will get a
4# and some 3# wins.
Because of the time it takes to educate people, most lotto
products won't tell you the advanced options you have to get
the most efficient coverage for your playing dollar. And yes,
the most likely return on a 3if wheel with or without multiple
wheels within wheels is some 3# tickets. And yes, in both
cases it is possible to get lucky and do better, but in the
case of wheels within wheels you put targets of known quality
in there on purpose. In the case of front loading, your
skills pay off when proven correct, rather then diluted across
a balanced wheel. You have to play in the way that's best for
you, provided you are aware you have choices. To learn more
about wheeling start here...
Philippe Coustaux's Wheeling: get OpenCover.zip the only thing
you have to remember to do is give your wheel a name ending in
.txt (ie: 46126.txt) so you can find it in the same directory.