Lottery wheeling

Lottery wheeling (also known as lottery system, lottery wheel, lottery wheeling system) is used by individual players and syndicates to distribute a subset of the possible lottery numbers across multiple tickets to ensure that at least one of these tickets will contain a winning combination if several draws are in this subset. For example, in a pick 5 lottery, a lottery system with a subset of 9 selections and a "3 if 3" guarantee means that a 3-win payout is ensured when three of these 9 selections are among the five numbers drawn.

A lottery wheeling system acts as a single ticket in terms of a particular guarantee, but allows playing with a larger set of numbers than will be drawn in the lottery. In contrast, a single ticket in a pick 6 lottery guarantees a 4-win if four of the player's numbers are drawn from a subset of only the six selections on that ticket. A lottery system with, say, 10 selections and the same guarantee would require at least 20 tickets but ensures a 4-win if four of the player's numbers are drawn from a subset of ten selections. The term "wheeling" comes from the cyclic permutation method of construction, as in the following example: a "pick 6, 9 numbers, 4 if 5" guarantee system in 3 combinations, where the chosen subset are the integers 1 to 9:

1.123456
2.456789
3.123789

This wheeling system defines three groups: A = {1 2 3]; B = {4 5 6}; and C = {7 8 9} and constructs the first ticket as AB, the second as BC and the third as CA. Any draw containing five of the numbers 1 through 9 will give at least one ticket with 4 of these numbers in it. We can confirm this by considering the possible distributions of the 5 numbers drawn among the three groups, and observing that there are always two groups that contain either all 5 or 4 of the 5 numbers drawn. Since any two groups are combined in a ticket, there will always be a 4-win or a 5-win. Difficulties greatly increase in constructing systems with more numbers and combinations. In mathematics, the study of these objects falls within the branch of combinatorial design. A lottery wheeling system has a basic guarantee (as in the examples above), but can have other, secondary guarantees.[1][2]

Lottery Wheels were in use in Europe and imported to Canada and the US in the 1970s. Wheeling systems usually try to guarantee a minimum number of wins given that draws fall in the set of the player's numbers. From a mathematical standpoint 'wheeling' has no impact on the expected value of a given ticket, but reduces the payout variance across all tickets, compared to making independent random selections for each ticket (from the same subset of winning numbers). Smoothing out the wins in this way is popular in syndicates.[3] Wheeling can be bundled with lottery prediction software and other tools which supposedly improve the odds but are often based on the Gambler's Fallacy or plain misunderstanding or misrepresentation of statistics.[4]

Full Wheel

Full Wheel includes all combinations that can be generated from a set of numbers a player picks, and therefore guarantees a first tier prize if all of the drawn numbers are within the player's set of numbers; it also guarantees a number of lower tier prizes. The only drawback with full wheels is they become fairly expensive with increasing the size of the set of the player's chosen numbers. A player who wishes to play a full wheel with 10 numbers in a pick 6 lottery game will have to play 210 combinations, while a full wheel with 15 numbers in the same lottery will require 5005 combinations!

In a famous occurrence, a Polish-Irish businessman named Stefan Klincewicz bought up 80% of the 1,947,792 combinations available at the Irish Lottery. He and his associates paid less than one million Irish pounds while the jackpot stood at 1.7 million pounds. The syndicate did have a ticket with the winning numbers. However, so did two other players, and the jackpot was split three ways. With the "Match 4" and "Match 5" prizes, though, Klincewicz's syndicate made a small profit overall.[5]

Abbreviated Wheel

An Abbreviated Wheel is an economical alternative for a Full Wheel. Although an Abbreviated Wheel does not include all possible combinations of the chosen numbers, it still guarantees at least one winning ticket if some of the numbers drawn are within the player's selection of numbers.

The following is an example of an abbreviated lottery wheeling system for pick-6 with 10 numbers, 4 if 4 guarantee and the minimum possible number of combinations for that guarantee, 20. The original system is given as 20 combinations on the numbers from 1 to 10. The next table gives a possible selection of the player's numbers and his/her set of tickets, which are obtained after substituting the numbers 1-10 with the player's numbers.

Numbers in the original system: 1 2 3 4 5 6 7 8 9 10
The player's numbers (example): 3 7 12 14 18 22 29 33 40 46
Original system The player's set
of tickets:
1. 1 2 3 4 8 9 1. 3 7 12 14 33 40
2. 1 2 3 5 6 7 2. 3 7 12 18 22 29
3. 1 2 3 5 9 10 3. 3 7 12 18 40 46
4. 1 2 4 5 8 10 4. 3 7 14 18 33 46
5. 1 2 4 6 7 8 5. 3 7 14 22 29 33
6. 1 2 6 7 9 10 6. 3 7 22 29 40 46
7. 1 3 4 5 6 10 7. 3 12 14 18 22 46
8. 1 3 4 5 7 8 8. 3 12 14 18 29 33
9. 1 3 5 6 8 9 9. 3 12 18 22 33 40
10. 1 3 7 8 9 10 10. 3 12 29 33 40 46
11. 1 4 5 7 9 10 11. 3 14 18 29 40 46
12. 1 4 6 8 9 10 12. 3 14 22 33 40 46
13. 2 3 4 5 7 9 13. 7 12 14 18 29 40
14. 2 3 4 6 9 10 14. 7 12 14 22 40 46
15. 2 3 5 7 8 10 15. 7 12 18 29 33 46
16. 2 3 6 7 8 9 16. 7 12 22 29 33 40
17. 2 4 5 6 7 10 17. 7 14 18 22 29 46
18. 2 5 6 8 9 10 18. 7 18 22 33 40 46
19. 3 4 6 7 8 10 19. 12 14 22 29 33 46
20. 4 5 6 7 8 9 20. 14 18 22 29 33 40

This example can be used to illustrate the main guarantee of the chosen system (a 4-win if four of the 10 player's numbers are drawn): Suppose the numbers 7,12,29, and 40 are drawn (these are shaded in the player's tickets), then the system guarantees at least one 4-win, by design. Indeed, it is easy to check that this is so. In fact, in this particular case, the system gives two 4-wins (in tickets 13 and 16), and it also gives seven 3-wins (these can be found in tickets 1,2,3,6,10,14, and 15). The number of combinations in an Abbreviated Wheel is significantly smaller than the number of combinations in a Full Wheel on the same set of numbers. In the example above, the Abbreviated Wheel for pick-6 lottery with 10 numbers and 4 if 4 guarantee has 20 tickets. A full wheel with 10 numbers requires 210 combinations and has 6 if 6 guarantee.

Lottery wheeling systems have been used by lottery players throughout the world. Full and Abbreviated Wheels are the most popular among different types of lottery wheels. Many lotteries provide the option of playing a full wheel either on a regular type of ticket or on a specially designed one without the need to fill all of the combinations individually. Several European lottery corporations have gone a step further and have provided the option of playing abbreviated wheels from a preapproved selection, by using specially designed playing slips which refer to the chosen system by number and do not require filling the individual combinations of the system.

Filtered Wheel

Filters can further reduce the number of combinations in a Full or Abbreviated Wheel. For example, a filter can be set to remove combinations with all odd numbers, to balance the number of odd and even numbers within the combination, etc. This leads to losing the guarantee in a Full Wheel. In a case of an Abbreviated Wheel, a better quality software will try to avoid the guarantee violation or if this is not possible, report the problem. Filters can be grouped into two classes: line filters and group filters.

Line filters:

  • Consecutives: set max. amount of pairs, triplets and quartets within a combination
  • Decades: how to distribute numbers between decades
  • Delta: set difference between adjacent numbers
  • Even/Odd: set max. amount of even and/or odd numbers within a combination
  • Final digits: set maximum matching final digits (e.g. 2, 12, 42)
  • Key positions: set key numbers at key positions (e.g. number 10 is always 2nd in every combination)
  • Movement: movement around the reference combination
  • Sequences: set sequences distribution
  • Spacing: set compulsory space between adjacent numbers
  • Spread: set the difference between the highest and the lowest number

Group filters:

  • Key groups: group of numbers to be placed into every combination
  • Key numbers: number(s) to be placed into every combination (for Key Number Wheels)
  • Low/High: set low/high numbers ratio
  • Roots: set roots span (root is when you add a number's digits, e.g. root of 23=2+3=5 )
  • Primes: set min/max for prime numbers
  • Subset: define a smaller group of numbers to appear in every combination and set the min/max amount
  • Sums: set acceptable min/max sum of all numbers in a combination

Key Number Wheel

Key Number Wheel (or a Power Number Wheel) is a wheel in which one or more numbers (key numbers or power numbers) appear in every combination of the wheel.

Maximized number arrangement (MNA)

MNA is a wheel in which the target result is to cover as many different winning combinations as possible within the tickets purchased without repeating a single winning combination.

Example:

In a pick 6 draw with prizes for 3, 4, 5 and 6 hits the player has 42 chances to win per line played.

Combination Count Possible combinations
3 Balls 20 XXXOOO, XXOXOO, XXOOXO, XXOOOX, XOXXOO

XOXOXO, XOXOOX, XOOXXO, XOOXOX, XOOOXX

OXXXOO, OXXOXO, OXXOOX, OXOXXO, OXOXOX

OXOOXX, OOXXXO, OOXXOX, OOXOXX, OOOXXX

4 Balls 15 XXXXOO, XXXOXO, XXXOOX, XXOXXO, XXOXOX

XXOOXX, XOXXXO, XOXXOX, XOXOXX, XOOXXX

OXXXXO, OXXXOX, OXXOXX, OXOXXX, OOXXXX

5 Balls 6 XXXXXO, XXXXOX, XXXOXX, XXOXXX, XOXXXX

OXXXXX

6 Balls 1 XXXXXX

If a single line were to cost $1 to play then that single ticket gives the player (100 / 42) = 2.38c per possible winning combination within the bet.

For the pick 6 draw, a correctly configured MNA will always maintain the 2.38c per bet regardless of the number of tickets played.

The following wheel demonstrates a basic MNA

------------------------------ Start ------------------------------

01 06 07 10 12 15

02 05 09 11 12 13

03 04 08 10 12 14

------------------------------ End ------------------------------

3 Ball Combinations = 60 out of 60 possible within 3 tickets: 100.00%

4 Ball Combinations = 45 of 45 possible within 3 tickets: 100.00%

5 Ball Combinations = 18 of 18 possible within 3 tickets: 100.00%

6 Ball Combinations = 3 of 3 possible within 3 tickets: 100.00%

Total unique combinations = 126 of 126 possible unique combinations within 3 tickets: 100.00%

The new bet price divided by the unique combinations in the 3 line wheel = 300 / 126 = 2.38c per

Therefore, it can be said the MNA is designed to cover the maximum number of unique combinations within the tickets played providing the player with the best possible coverage for the money spent.

The result is an overall increase of prizes covered per play compared to wheels that inherently repeat combinations .

gollark: You can actually do `(nil).z` and right now it'll return 5 or something.
gollark: Also anything else at random.
gollark: PotatOS breaks a small minority of programs which do weird exotic things.
gollark: Default programs *will* work.
gollark: Stuff has to explicitly opt in to use of your APIs.

See also

References
  1. Iliya Bluskov, "Combinatorial Systems (Wheels) for Pick-5 lotteries, including Euromillions and the Mega lotteries", Lotbook Publishing, 2011.
  2. Iliya Bluskov, "Combinatorial Systems (Wheels) for Pick-6 lotteries", Lotbook Publishing, 2012.
  3. "What is a Lottery Wheeling System?"
  4. Leah Brown, "An Analysis of Lottery Master Guide", Brigham Young University, 2003.
  5. Rebecca Fowler, "Irish syndicate reveals the painstaking method for beating the odds", The Independent, 1996.
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