Shortened and Full Systems — What's the Difference

Both solutions address the same problem: a player has a hunch about eight, ten, or twelve numbers, but the coupon only allows six to be marked. They differ in how they solve that problem — and in the math behind it. Below, first the numbers, then an explanation of where they come from.

How many bets and what they cost — table

The number-of-bets column is the value of the binomial coefficient: how many different six-number combinations can be formed from the given pool. Cost is calculated at the price of 3 PLN per bet in Lotto. The last column shows how many times more expensive each subsequent tier is compared to the previous one.

Numbers in pool	Bets (full system)	Cost at 3 PLN	Increase vs. previous
7	7	21 PLN	—
8	28	84 PLN	4×
9	84	252 PLN	3×
10	210	630 PLN	2.5×
12	924	2,772 PLN	4.4× vs. 10
15	5,005	15,015 PLN	5.4× vs. 12
20	38,760	116,280 PLN	7.7× vs. 15

The number of bets is the six-element combination from pool n, i.e. the binomial coefficient n choose 6. Prices assume a rate of 3 PLN per single Lotto bet — playing across several draws or adding Lotto Plus increases the cost proportionally.

Where the number of bets in a full system comes from

A full system doesn't require any specialist knowledge — it's simply writing out every six-number combination from the chosen pool. The number of such combinations is given by the binomial coefficient: for a pool of 10 numbers, that's 10! divided by 6! times 4!, which equals 210. Each of these 210 bets must be paid for separately, because each is a separate selection on the coupon.

The guarantee here is maximal and very simple to describe: if all six drawn numbers are within our pool, then one of the bets is a full six-number match — by definition, because we wrote out every possible combination. Matching five numbers from the pool guarantees a full set of fours and fives; matching four numbers guarantees a full set of threes and fours. The math is predictable down to a single bet, it just grows faster than most players expect.

This is visible in the last column of the table. Adding just one number to a ten-number pool costs more than the entire previous system, and with a pool of fifteen numbers we're already talking about an amount comparable to the price of a used car. This is the main reason shortened systems were created. You can find ready-made tables of combinations for various pool sizes on the shortened and full systems page.

What exactly a shortened system guarantees

A shortened system is a subset of bets from the full system, chosen to satisfy a specific condition. The condition is usually written as: with a pool of n numbers, if we hit k of the drawn numbers within it, we guarantee at least one match of tier m. Example: a ten-number system with a guarantee of four matches when five numbers are hit means that if five out of the six drawn numbers are within our ten-number pool, at least one bet contains four of them.

Conditionality is key here and most often overlooked. The guarantee only works if the assumed hit in the pool actually occurs. If only three out of the six drawn numbers land in our ten-number pool, the shortened system promises nothing — just as a full system doesn't promise a six-number match if one of the drawn numbers wasn't selected by us at all.

Constructing such sets is a problem in combinatorics, closely related to covering design theory. For many parameter combinations, the optimal solution isn't known, so published systems are the best sets found so far, not always the smallest possible ones. This is why different sources sometimes give a different number of bets for the same guarantee.

What no system changes

The probability of hitting a six-number match in Lotto is one in 13,983,816 for a single bet. Playing 210 bets raises it to 210 in 13,983,816, or about one in 66,590 — exactly what would result from playing 210 random, different bets without any system. A system doesn't change the odds per złoty spent; it changes the distribution of outcomes.

This is its entire role. A hundred random bets produce scattered low-tier matches or none at all; a hundred bets arranged systematically from one pool produce a concentrated result: either nothing, or many matches at once, if the drawn numbers fell within our pool. The total payout over the long run remains the same, because payouts are proportional to the number of winning bets.

There's also no way to choose numbers for the pool that would increase the chance those particular numbers will be drawn. Each draw is independent of previous ones, and past frequency of occurrence has no effect on the outcome of the next draw. You can browse historical distributions in the Lotto statistics, but they're for looking at history, not for predicting the future.

Who should use a full system, who should use a shortened one

A full system makes sense with a small pool — seven or eight numbers, meaning 21 or 84 PLN. The guarantee is then complete and easy to understand, and the cost fits within a budget you might otherwise spend on regular bets over several draws. Above ten numbers, the math stops making sense for a single player.

A shortened system is the answer for a pool of twelve, fifteen, or twenty numbers, where a full system would cost thousands. However, you need to know what you're signing up for: a reduced guarantee means that even with a full hit on the pool, you might not get a six-number match, only a handful of four-number ones. It's a conscious trade-off of top-tier certainty for price.

• Pool of 7–8 numbers: full system, cost 21–84 PLN, maximum guarantee
• Pool of 9–10 numbers: full system 252–630 PLN, or shortened with a four-match guarantee for a fraction of that amount
• Pool of 12+ numbers: practically only shortened systems, full system is 2,772 PLN and up
• Group play: a full system from a larger pool split among several participants

In group play, what matters most isn't the math but the arrangements made on paper: who pays for the bets, who keeps the coupons, and in what proportions any winnings will be shared. It's also worth remembering that winnings above 2,280 PLN are subject to a ten-percent flat-rate tax — details are covered in our article on the tax on winnings.

The most common misunderstandings

First: that a system increases the chance of winning. It increases it only by as much as follows from the number of bets paid for — exactly the same as buying that same number of regular coupons. Second: that a guarantee in a shortened system means a certain payout. It only means matching a certain tier, on the condition that the assumed number of drawn numbers fell within our pool.

Third: that a guaranteed four-number match will cover the cost of the system. A fourth-tier win in Lotto is usually a few dozen złoty, while a shortened system from a twelve-number pool costs dozens of bets. Fourth: that systems for different games are interchangeable. They're not — the number pool and the number of selections differ in each game, so a set built for Lotto doesn't apply to Eurojackpot or Mini Lotto.