Lottery Wheel: A Clever Mathematical Strategy That Works


Last updated on April 1, 2024

A lottery wheel is a covering strategy designed to trap the winning numbers. Believe it or not, this strategy cleverly works.

The only logical way to increase your chances of winning the lottery is to increase the number of tickets you play.

However, simply purchasing additional tickets at random is unlikely to be successful. Buying many tickets is more effective when done strategically.

Fortunately, mathematics has a nice version of that strategic method. We call it “covering.” And that’s how a lottery wheel comes in handy.

Does that pique your interest? If so, let’s dig in.

Lottery Wheel and Combinatorics

Covering is so powerful that you don’t have to worry about hot and cold numbers. And you can even say goodbye to lucky and unlucky numbers (if those numbers exist).

In the lottery, numbers are just symbols to represent individual balls.

Instead of numbers, balls can be represented by animals, fruits, or objects. So it doesn’t matter what symbols you like to pick.

So, in a lottery game, we deal with this selection of objects under combinatorics.

What is combinatorics?

Combinatorics1 is a branch of mathematics that deals with the combination and permutation2 of objects belonging to a finite set of elements and the mathematical relations that characterize their properties.3

The many different ways of combining elements in a set are among the main tasks in combinatorial design.4 Combinatorial design can be useful in various practical applications, from statistics, computer science, and business to economics.

In the world of lottery games, we take advantage of combinatorics by using a lottery wheel.

There are many kinds of lottery wheels. Let’s talk about the most popular ones below.

Full Wheel

The wheeling system allows you to pick more numbers, increasing your chances of matching the numbers drawn in a lottery draw.

For instance, in a lotto 6/49 game, selecting ten numbers will produce 210 possible combinations for you.

Suppose you pick 6,8,14,15,21,26,27,41,42,44 and the numbers 8, 15, 27, and 44 are drawn, then the system provides you with the following matches:

Here are examples of a lottery wheel using 10 numbers as a covering size. We assume that 4 winning numbers belong to the original 10 selections. Here are the possible outcomes:
15 counts of 4-matches, 80 counts of 3-matches, and 90 counts of 2-matches. This demonstrates a lottery formula based on lottery wheel.

There is one caveat, though. You only win when some of the winning numbers are among your selections. Otherwise, you don’t win anything.

The disadvantage of the full-wheeling system is that it tends to become expensive since it must produce all the possible lines. The more numbers you select, the more combinations you need to buy for maximum win coverage.

For instance, selecting 11 numbers for a lotto 6/49 will produce 462 possible combinations. If you pick 12, then the combinations will quickly increase to 924.

Abbreviated Wheel

An abbreviated wheel is the reduced version of a full-wheeling system. It does not include all the possible combinations out of your selection. Hence, this wheeling system is said to be an economical alternative.

In a full-wheeling system, selecting ten numbers produces 210 possible combinations. However, in an abbreviated system, ten numbers only produce ten lines.

But you win fewer matches because an abbreviated system does not include all the possible combinations. Nonetheless, it guarantees at least one win if some of the winning numbers are among your chosen numbers.

If we use the same set of numbers 6,8,14,15,21,26,27,41,42,44, then below is an example of a reduced list of combinations using an abbreviated system:

This abbreviated lottery wheel produced 10 lines of combinations rather than the full 210 combinations.

If 8, 15, 27, and 44 are drawn, then the system provides you with the following matches:

With the abbreviated lottery wheel, you get only a few matches. Here are the possible outcomes for the 10 numbers, assuming that four numbers were drawn correctly: zero 4-matches, four 3-matches, and six 2-matches.

Which Lottery Wheel Works Better?

Let’s say we pick these ten numbers 3,8,13,16,21,22,35,39,40,46. The table below will show you how many combinations are possible for each wheeling system.

Full lottery wheel system will produce 210 lines of combinations. An abbreviated wheel will produce 10 lines.

We will assume four scenarios to see the huge difference between the two systems. And for each scenario, we will count how many ways you get winning matches for each system.

First scenario: If 8, 35, and 40 were drawn (3 of 10 numbers)

In a lottery wheel using abbreviated system, you get 1 count of 3 matches. In full-wheel system, you get 35 counts of 3-matches

You may not see the big difference in this scenario. For most lottery systems, getting three correct numbers is not too exciting.

Let’s see what will happen if we get four numbers correct.

Second scenario: If 8, 21, 35, and 40 were drawn (4 of 10 numbers)

You get 4 counts of 3-matches in the abbreviated system versus 80 counts from the full lottery wheel.

As shown in the table above, you can see a big loss from not having four matches in the abbreviated version. In the full version, having four matches 15 times is quite a consolation, and the many 3-matches are not bad.

Third scenario: If 8, 13, 21, 35, and 40 were drawn (5 of 10 numbers)

In abbreviated system, you get 10 counts of 3-matches, nothing for 4-matches, nothing for 5-matches. In the full lottery wheel, you get 99 counts of 3-matches, 51 counts of 4-matches, and 5 counts of 5-matches.

When you get five winning matches from your selection, the full version will surely give you that guarantee.

Fourth scenario: If 8, 13, 21, 35, 40, and 46 were drawn (6 of 10 numbers)

You get 6 counts of 4-matches and 4 counts of 3-matches in the abbreviated system. In the full-system, you get 80 counts of 3-matches, 90 counts of 4-matches, 24 counts of 5-matches, and 1 count of a jackpot combination.

And here is where the big difference shows. The table above tells that the full wheel gives you all the reasons to celebrate.

Based on the tables above, we can arrive at the following conclusions:

Abbreviated SystemFull-wheel System
You get the guarantee that you will win the jackpot if six numbers from your selections were drawn.You get the guarantee that you will win the jackpot if six numbers from your selections are drawn.

Now, which one is better? The answer is neither.

Let me explain.

The Truth About Lottery Wheel

You’ve probably heard or read that the way to win fast is to stop hitting the jackpot and earn small prizes to keep you in the playing loop until you win.


There is something you need to know about why an ordinary lottery wheel doesn’t work.

An abbreviated system doesn’t produce the best group of combinations

Lotto players like to play with a system that guarantees sure winning at a minimal cost. So, the abbreviated system is the solution to the problem.

Or is it? I’ll let you be the judge.

Let’s use the same set of numbers from the previous section:

Here is a set of 10 numbers: 3,8,13,16, 21,22,35,39,40,46. We will subject these numbers in an abbreviated lottery wheel.

To get a minimal list of combinations, we can divide the numbers into five groups:

the numbers are grouped into 5 letters. 3 and 8 belongs to group A. 13 and 16 belongs to group B. 21 and 22 belong to group C. 35 and 39 belong to group D. 40 and 46 belong to group D.

Out of this grouping scheme, we can produce ten lines, as shown in the table below:

The abbreviated wheel can produce the following 10 lines: a-b-c, a-b-d, a-b-e, a-c-d, a-c-e, a-d-e, b-c-d, b-c-e, b-d-e, c-d-e

Let’s determine how likely you will win the jackpot using this abbreviated wheeling method. Of course, the common lottery rule says you must match the six winning numbers to win the jackpot. So, to make it happen, your abbreviated wheel should look like this:

A-B-C pattern = 3-8-13-16-21-22

That is if the winning numbers are 3-8-13-16-21-22.

Or like this:

A-C-E pattern = 3-8-21-22-40-46

That is if the winning numbers are 3-8-21-22-40-46.

Or like this:

B-D-E pattern = 13-16-35-39-40-46

That is if the winning numbers are 13-16-35-39-40-46

However, the lottery has a different plan. In other words, a random game doesn’t work that way.

Let’s prove that from the actual lottery draws. Let’s use the Canada Lotto 6/49. You can download the full list of results from June 1982 up to the current year from the official website of the Canada Lotto 6/49.

At the time of writing, my dataset is from June 1982 to September 2018. So that’s a total of 3,688 draws in 36 years.

My study shows that trapping five numbers from our ten selections occurred six times in 36 years. Therefore, you should be able to get some five-number matches six times, right?

Not exactly.

The chances that you get 5-number-matches from the reduced ten lines are unlikely. Let’s look at those draws where five winning numbers are found in the original ten numbers.

Draw No. 580
Draw Date: 1989-08-12
Winning Numbers: 03-04-13-21-22-46

A-B-C-E pattern = 3 from A, 13 from B, 21-22 from C, 46 from E

Composition: 2 numbers from box C, 1 number from box A, 1 number from box B, and 1 number from box E.

line 1 = 3-8-13-16-21-22 (4 matches), line 3 = 3-8-13-16-40-46 (3 matches), line 4 = 3-8-21-22-35-39 (3 matches), line 5 = 3-8-21-22-40-46 (4 matches), line 7 = 13-16-21-22-35-39 (3 matches), line 8 = 13-16-21-22-40-46 (4 matches), line 10 = 21-22-35-39-40-46 (3 matches).

So if five winning numbers are within your selection of 10 numbers, your winning lines may compose 4 and 3 matches, but you don’t get five matches.

Draw No. 635
Draw Date: 1990-02-21
Winning Numbers: 13-16-21-26-35-46

B-C-D-E PATTERN = 13-16 from B, 21 from C, 35 from D, 46 from E

Composition: 2 numbers from box B, 1 number from box C, 1 number from box D, and 1 number from box E.

4 lines of 3-matches. 3 lines of 4-matches.

Based on the above winning numbers, you don’t get 5-number-matches, only 3-matches, and 4-matches.

Draw No. 1202
Draw Date: 1995-07-29
Winning Numbers: 08-13-16-37-39-46

A-B-D-E = 8 from A, 13-16 from B, 39 from D, and 46 from E.

Composition: 2 numbers from box B, 1 number from box A, 1 number from box D, and 1 number from box E.

4 lines of 3-matches. 3 lines of 4-matches.

Time and again, a random game will never follow your abbreviated system. Your winning lines will consist only of 4 and 3 matches.

Draw No. 2347
Draw Date: 2006-07-19
Winning Numbers: 08-13-21-22-24-39

A-B-C-D = 8 from A, 13 from B, 21-22 from C, 39 from D

Composition: 2 numbers from box C, 1 number from box A, 1 number from box B, and 1 number from box D.

4 lines of 3-matches. 3 lines of 4-matches

We can continue with the results; you’ll never find any five matches.

Draw No. 2769
Draw Date: 2010-08-04
Winning Numbers: 05-08-13-22-35-40

A-B-C-D-E = 8 from A, 13 from B, 22 from C, 35 from D, and 40 from E

Composition: 1 number from box A, 1 number from box B, 1 number from box C, 1 number from box D, and 1 number from box E.

10 lines of 3-matches. No 4-match

Draw No. 3093
Draw Date: 2013-09-11
Winning Numbers: 13-21-22-26-35-40

B-C-D-E = 13 from B, 21-22 from C, 35 from D, and 40 from E.

Composition: 2 numbers from box C, 1 number from box B, 1 number from box D, and 1 number from box E.

4 lines of 3-matches. 3 lines of 4-matches.

What is the message?

Based on the above analysis, we can notice one particular observation. We can easily describe this observation using the image below:

This image describes what is unlikely thing to happen in a lottery wheel. Matching 4 numbers from 2 boxes is an unlikely event to occur. Matching 6 numbers from 3 boxes is an unlikely event to occur. For example A-C-E is an unlikely pattern to occur. Therefore 3-8-21-22-40-46 is unlikely to occur as a winning combination.

In a random 6/49 game, an event such as capturing all the six winning numbers in 3 boxes is an improbable thing to happen. In fact, in 36 years of Canada Lotto 6/49, such an event has never occurred for our 10-number selection.

The reason for this is simple. According to probability theory, a random lottery game distributes the probability fairly and evenly across the entire number field. The same rules apply to all lottery systems worldwide.

But don’t get me wrong. An event such as trapping the six winning numbers in three boxes is still possible. We are saying that it is improbable because the lottery is subordinate to the dictates of probability theory.

Why Is The Abbreviated Wheel Ineffective?

Well, the success-to-failure ratio has everything to do with it.

Our analysis starts with the available sets. Our sets should look like this:

A = {3, 8}

B = {13, 16}

C = {21, 22}

D = {35, 39}

E = {40, 46}

Using combinatorics, we can determine all the possible compositions and separate them into combinatorial groups with varying success-to-failure ratios.

We divide the compositions into three groups as follows:

CompositionsS/F RatioCategory
Group 1
Composition #1, #2, #3, #4, #5
Group 2
Composition #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35
Group 3 (This is the abbreviated system)
Composition #36, #37, #38, #39, #40, #41, #42, #43, #44, #45

Combinations produced using the abbreviated system fall under group 3. In Lotterycodex calculations, this group represents compositions that rarely occur in a lottery draw. Hence, they are improbable.

The table explains why the abbreviated system doesn’t work well. If you plan to play ten lines using the abbreviated system in 1000 draws and pay $2 per ticket, you will spend $20,000 on tickets, and all you will get are 3 and 4 matches. It’s not a worthwhile exercise altogether.

No doubt, compositions #1, #2, #3, #4, and #5 are the dominant compositions. The actual results of the Canada Lotto 6/49 game speak for it. See the tables below:

1989-08-1203-04-13-21-22-46#1 (found twice),
#10, #32, #32
None from group 3
1990-02-2113-16-21-26-35-46#1 (found twice),
#10, #32, #32
None from Group 3
1995-07-2908-13-16-37-39-46#5 (found twice),
#11, #27, #28,
None from Group 3
2006-07-1908-13-21-22-24-39#1 (found twice),
#12, #33, #34
None from Group 3
2010-08-0405-08-13-22-35-40#1, #2, #3, #4, #5
None from group 2
None from Group 3
2013-09-1113-21-22-26-35-40#1 (found twice),
#15, #17, #19
None from Group 3

Notice that compositions from the abbreviated group never produced five matches.

It’s easy to see that your strategy should focus on the dominant compositions. However, determining these dominant compositions can be a mathematical challenge, especially when dealing with large numbers.

It would be best to have a lottery wheel that can handle complex combinatorial calculations. We created the Lotterycodex calculator to do just that.

Lotterycodex as a Lottery Wheel

Lotterycodex offers you an advanced lottery wheel that goes beyond the basic. The calculator generates the list of possible combinations from your covering size and separates combinatorial groups according to success-to-failure ratios.

Below is the results of a Lotterycodex analysis for a 5/50 game:

If you’re implementing a wheeling strategy when playing the lottery, Lotterycodex suggests focusing on the dominant groups. These are the groups with favorable success-to-failure ratios.5

If you are interested in the nitty-gritty aspect of Lotterycodex calculations, I invite you to read The Winning Lottery Formula Using Math. We don’t hide the formula.

Questions and Answers

What is a lottery wheel?

A lottery wheel is a strategic system used in lottery games to select numbers to enhance the odds of winning. It involves arranging numbers into various combinations based on combinatorial principles. This strategy aims to trap winning numbers more strategically than random selection. It’s designed to increase the chances of matching some or all of the winning numbers, thereby improving the likelihood of winning lower-tier prizes, even securing a jackpot win.

How does Lotterycodex approach lottery wheel strategies?

Lotterycodex focuses on optimizing the selection of number combinations by analyzing success-to-failure ratios. This method involves organizing the numbers into a wheel system and separating combinatorial groups into dominant, occasional, and rare ones, helping lottery players make informed decisions when selecting combinations.

Is the lottery wheel strategy more effective than random number selection?

The lottery wheel strategy is more effective than random number selection because it strategically traps winning numbers. This method improves the probability of winning by covering a wide range of possible combinations based on a set of numbers.

Additional Resources

  1. Basics of Combinatorics    []
  2. Brilliant: Combinatorics    []
  3. Wolfram Mathworld: Combinatorics    []
  4. Combinatorial design    []
  5. The Lotto Secret: Three Math Strategies for Winning Revealed    []
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