Let's talk about wheeling systems — partial number systems used in lottery play. How can a system like this actually help a player? Why are some systems called "full" and others "partial"? And what's the advantage of a wheeling system?
Full wheel systems
Full wheel systems are considered excessive and generally aren't cost-effective. That said, there's a hypothetical scenario where a full system can pay off. Say, in a 6-from-45 lottery, you've picked 10 numbers and you're absolutely convinced all 6 winning numbers are somewhere in that set of 10. You'd lay out those ten numbers in a full system. Full systems can genuinely pay off when a jackpot is large, since your odds of landing it depend directly on how many combinations you're covering, not on how "wasteful" the system is. In that scenario, the smaller losses that come from lower odds on second- and third-tier prizes don't really matter. And if you're dealing with fixed-odds bets, the inefficiency of a full system matters even less, making full systems a genuinely reasonable choice.
So what exactly is a full system? It's a system built from every possible combination of a chosen set of numbers. The number of combinations in a full system can be calculated using the binomial coefficient (Bn):
Bn = N * (N - 1) * (N - 2) * … * (N - (K - 1)) / K!
For a "6 of 45" lottery, the binomial coefficient works out as:
Bn = (45 * 44 * 43 * 42 * 41 * 40) / (1 * 2 * 3 * 4 * 5 * 6) = 8,145,060.
The big downside of full systems is the cost — it adds up fast.
Wheeling (partial) systems
To cut down on the cost of playing, a large number of wheeling (partial) systems have been developed over the years.
When you've got a large pool of numbers to choose from and want to maximize your odds of winning, it's usually better to use a partial system with minimal redundancy. That approach makes the most sense when a lottery's jackpot is relatively modest.
So what exactly is a wheeling system? It's a system built from a defined set of numbers, designed to give you a shot at winning any prize tier below the jackpot, as long as the system contains all the actual winning numbers. That said, wheeling systems don't rule out winning the jackpot either — they just aren't specifically built to guarantee it.
What goes into building a wheeling system? First and foremost, it's built by comparing two full systems. Let's go back to our "6 of 45" example, and say we want a wheeling system covering 20 numbers with a guaranteed "3-number match." To build it, we compare two full systems: 45/6 and 20/3. The number of possible combinations (Bn) works out to 8,145,060 for the first system, and just 1,140 for the second. We've found what we're looking for once every "3-number" combination from the 20/3 system fits inside a "6-number" combination from the 45/6 system. Where combinations overlap, the system needs to be optimized by removing the duplicates.
Working through that much data by hand is extremely tedious and time-consuming. That's exactly why various automated tools exist to calculate the most likely winning combinations. The benefit is obvious: using software eliminates the risk of the calculation errors that are all too common when this is done by hand.
Advantages of wheeling systems
So what's the actual point of a wheeling system? It's simple: your odds of matching a winning 6-number combination from a pool of 20 numbers are dramatically better if you run the exact math, compared to just guessing at a combination and hoping for the best.
If all 6 winning numbers happen to fall within your 20-number pool, your odds of winning a "3-number match" prize come out to 100%.
If you match 5 numbers, your odds of winning come to 80%; matching 4 correctly gives you 43%; and matching 3 gives you 14%. The odds of landing a "4-number match" prize when you've correctly chosen all 6 numbers works out to around 30%, and the odds of a "5-number match" come to about 2% (so that outcome is possible too).
The second major advantage of wheeling systems over full systems is the noticeably lower cost of play. The example above covers a system guaranteeing a "3-number match." But plenty of other systems exist offering even stronger guarantees.
In practice, any wheeling system that suits your playing style works about as well as any other, whatever guarantee level you choose. There are so many ready-made partial systems out there that it's easy to find one that fits what you're after.
The takeaway: wheeling systems have a clear edge, and are really the only genuinely effective way to calculate winning combinations that can be applied to almost any lottery play.
Examples of wheeling systems in action
Let's walk through one of the simplest wheeling systems for a "6 of N" lottery — one that guarantees a "2-number match" if you've correctly picked at least two of the winning numbers.
The "7 numbers - 3 lines" system
Here's how it works. All you need to do is pick seven numbers to play and map them to the corresponding positions. Here's an example mapping: 1=>3, 2=>7, 3=>11, 4=>29, 5=>33, 6=>40, 7=>43. Here's what that produces:
| 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 2 | 3 | 4 | 5 | 7 |
| 1 | 2 | 3 | 4 | 6 | 7 |
| 3 | 7 | 11 | 29 | 33 | 40 |
| 3 | 7 | 11 | 29 | 33 | 43 |
| 3 | 7 | 11 | 29 | 40 | 43 |
This system guarantees:
- if you match 3: 0-3 "3-number matches"
- if you match 4: 3 "3-number matches," or 1-3 "4-number matches" + 0-2 "3-number matches"
- if you match 5: 3 "4-number matches," or 1-2 "5-number matches" + 1-2 "4-number matches"
- if you match 6: 3 "5-number matches," or 1 "6-number match" + 2 "5-number matches"
The "14 numbers - 25 lines" system
For this system, we'll use the following numbers: 1=>1, 2=>2, 3=>4, 4=>8, 5=>11, 6=>14, 7=>21, 8=>24, 9=>26, 10=>29, 11=>31, 12=>35, 13=>38, 14=>39.
| 1 | 3 | 10 | 11 | 13 | 14 |
| 1 | 3 | 9 | 12 | 13 | 14 |
| 1 | 2 | 3 | 4 | 13 | 14 |
| 2 | 5 | 6 | 9 | 10 | 14 |
| 1 | 3 | 6 | 7 | 13 | 14 |
| 1 | 3 | 5 | 8 | 13 | 14 |
| 2 | 7 | 8 | 11 | 12 | 14 |
| 1 | 3 | 5 | 7 | 9 | 11 |
| 4 | 6 | 8 | 10 | 12 | 14 |
| 4 | 5 | 7 | 9 | 11 | 14 |
| 1 | 3 | 6 | 8 | 10 | 12 |
| 4 | 5 | 8 | 9 | 12 | 13 |
| 2 | 9 | 10 | 11 | 12 | 13 |
| 2 | 5 | 6 | 7 | 8 | 13 |
| 4 | 6 | 7 | 10 | 11 | 13 |
| 1 | 2 | 3 | 4 | 11 | 12 |
| 1 | 2 | 3 | 4 | 9 | 10 |
| 1 | 2 | 3 | 4 | 7 | 8 |
| 2 | 5 | 6 | 11 | 12 | 14 |
| 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 7 | 8 | 9 | 10 | 14 |
| 1 | 3 | 5 | 7 | 10 | 12 |
| 1 | 3 | 6 | 8 | 9 | 11 |
| 4 | 6 | 7 | 9 | 12 | 13 |
| 4 | 5 | 8 | 10 | 11 | 13 |
| 1 | 4 | 29 | 31 | 38 | 39 |
| 1 | 4 | 26 | 35 | 38 | 39 |
| 1 | 2 | 4 | 8 | 38 | 39 |
| 2 | 11 | 14 | 26 | 29 | 39 |
| 1 | 4 | 14 | 21 | 38 | 39 |
| 1 | 4 | 11 | 24 | 38 | 39 |
| 2 | 21 | 24 | 31 | 35 | 39 |
| 1 | 4 | 11 | 21 | 26 | 31 |
| 8 | 14 | 24 | 29 | 35 | 39 |
| 8 | 11 | 21 | 26 | 31 | 39 |
| 1 | 4 | 14 | 24 | 29 | 35 |
| 8 | 11 | 24 | 26 | 35 | 38 |
| 2 | 26 | 29 | 31 | 35 | 38 |
| 2 | 11 | 14 | 21 | 24 | 38 |
| 8 | 14 | 21 | 29 | 31 | 38 |
| 1 | 2 | 4 | 8 | 31 | 35 |
| 1 | 2 | 4 | 8 | 26 | 29 |
| 1 | 2 | 4 | 8 | 21 | 24 |
| 2 | 11 | 14 | 31 | 35 | 39 |
| 1 | 2 | 4 | 8 | 11 | 14 |
| 2 | 21 | 24 | 26 | 29 | 39 |
| 1 | 4 | 11 | 21 | 29 | 35 |
| 1 | 4 | 14 | 24 | 26 | 31 |
| 8 | 14 | 21 | 26 | 35 | 38 |
| 8 | 11 | 24 | 29 | 31 | 38 |
This system guarantees:
- if you match 3: 1-4 "3-number matches"
- if you match 4: 4-7 "3-number matches," or 1-2 "4-number matches" + 0-5 "3-number matches"
- if you match 5: 10-14 "3-number matches," or 1-4 "4-number matches" + 4-12 "3-number matches," or 1 "5-number match" + 0-2 "4-number matches" + 3-10 "3-number matches"
- if you match 6: 2-8 "4-number matches" + 2-17 "3-number matches," or 1-2 "5-number matches" + 1-5 "4-number matches" + 5-14 "3-number matches," or 1 "6-number match" + 0-3 "4-number matches" + 7-16 "3-number matches"
Give it a try! And good luck out there. We'll keep adding new wheeling systems over time, so you can experiment, play, and win.