# Lottery wheeling systems

Abbreviated Wheeling is a lottery playing system that doesn’t require unfolding all combinations from a player’s number set, yet it still guarantees at least one winning variant contained in the resulting unfoldment, provided a certain amount of drawn numbers fall within the set.

Abbreviated Wheels contain the mathematically minimum number of criteria-meeting combinations to guarantee a win with a predetermined amount of number. It’s a good strategy to keep expenses balanced when playing multiple lottery tickets or unfolded combinations.

Suppose you are going to play a lottery with the matrix 5/39 and you have selected 9 numbers into your number set – 2, 5, 9, 17, 20, 26, 27, 33, 39, for instance. You then specify a criteria, such as “3-win”. This means, any three numbers drawn must fall within your set to guarantee the criteria. An Abbreviate Wheel for all “three via five of nine” is suitable, and, it has the minimum possible of 12 combinations. See the illustration below.

##### How Are Full Wheels Different?

What about the Full Wheel? Let’s count all the different five-number combinations that can be unfolded from a nine-number set? There are 126 combinations. Too many, isn’t it? That would be quite expensive to purchase lottery tickets with the Full Wheel. However, the Full Wheel for the items above guarantees 15 combinations with 3-wins, as long as any three numbers in the set match the drawn numbers. And so on… Five the 4-win if four of the numbers drawn match; one the 5-win if five of the numbers match. As we can see, there are fewer combinations in our Abbreviated Wheel than a Full Wheel with the same number set.

Abbreviated Wheels and Full Wheels are the most commonly used types of lottery systems worldwide. Some lotteries allow participants to pick more numbers in one field of a ticket without needing to fill each unfolded combination separately. This is known as a wide-sized combination. Any wide-sized combination by default forms a Full Wheel automatically when it’s envisaged by a lottery. But only a few lotteries have automated tools for Abbreviated Wheeling applications which provide the prerequisite covers for tickets. That’s why DigitWheel was created.

Some lottery players dedicate a lot of time hand writing unfolded combinations of their number sets on paper, while others, particularly those with more advanced combinatorial skills, will try to draw an incidence matrix and then search combinations corresponding to the rows that form a covering. This method lets them find a covering, but they have to ensure that there are no other coverings with fewer entries (i.e. they have to look for the minimum number of covered rows). This not easy to do, and is something that’s closely connected with the ideas of Paul Erdős’ conjecture and hypergraph Turán problem.

The main condition under which an Abbreviated Wheel allows you to win is that your set has to have enough numbers coinciding with the lottery drawing. This is all luck. Nothing you can do will influence the drawing. So always keep that in mind when playing. Using an Abbreviated Wheel is a good strategy that allows you to balance expenses when more numbers are engaged and unfold them into simple combinations. However, wins are not 100% guaranteed.

Okay, so let’s see how the Abbreviated Wheel system (above) works when all test numbers are fully randomized, just like in a real lottery drawing. Suppose the lottery numbers drawn are (randomize it again!). In this test, there are a maximum of numbers matched in any combination above.