Twenty-seven tickets.
A guaranteed win. A £54 loss.

The one theorem in the building

Every other page in this playground has been an argument that a pattern is not there. This one has a proof that something is.

A covering design $C(v, k, t)$ is a collection of $k$-element blocks drawn from a $v$-element set, arranged so that every $t$-element subset lies inside at least one block. A lotto design $L(v, k, p, t)$ is the weaker, more useful cousin: choose $v$ numbers, buy tickets of size $k$, and demand that whenever $p$ of your $v$ numbers are drawn, some ticket matches at least $t$ of them.

That is a conditional, and the conditional is the whole content:

$$\text{IF } |D \cap V| \ge p \text{ THEN } \exists\, T \in \mathcal{B} : |T \cap D| \ge t$$

There is no probability anywhere in it. It is a statement about what the arrangement makes impossible, and it is either true or false of a given set of tickets — checkable, in principle, by looking at every draw.

The Fano plane

The smallest object that does this job properly is the projective plane of order 2: seven points, seven lines, three points on every line, and every pair of points on exactly one line.

Count it. There are $\binom{7}{2} = 21$ pairs of points. Each line contains $\binom{3}{2} = 3$ pairs. Seven lines times three pairs is twenty-one, and no pair is covered twice — the plane covers everything with not one block to spare. Perfection of this kind is rare enough in combinatorics to be worth a moment.

Now the move that turns it into a lottery wheel. Let each point stand for a pair of numbers rather than a single one. A line is three points, so a line is six numbers, so a line is exactly one ticket. Fourteen numbers, seven tickets — and because every two points share a line, any two of those fourteen numbers appear together on some ticket.

Three planes and two triangles

Six balls are drawn. Split the 59 numbers into five groups; by pigeonhole, some group must contain at least two of the six. So if every group has its pairs covered internally, a 2-match is unavoidable.

Three Fano planes, each covering 14 numbers with 7 tickets, handle 42 of them. For the rest, a triangle does the same trick more cheaply: three points, three edges, every pair of points on an edge. Let the points be triples of numbers, and an edge — two points — is again six numbers, one ticket. Nine numbers, three tickets.

$$\underbrace{14 + 14 + 14}_{\text{three Fano planes}} + \underbrace{9 + 8}_{\text{two triangles}} = 59, \qquad \underbrace{7 + 7 + 7}_{} + \underbrace{3 + 3}_{} = 27$$

Twenty-seven tickets, and no draw in the 45,057,474 that a 6-from-59 game can produce escapes them. Cushing and Stewart also showed that 26 cannot be made to work, by exhaustive constraint search — so 27 is not merely sufficient, it is the answer.

Why finding these is hard

The covering number $C(v,k,t)$ — the fewest blocks that will do — is known exactly only for scattered small cases. The general problem is a hard combinatorial search, and the field's living record is the La Jolla Covering Repository, a decades-long communal archive of the best constructions anyone has found.

The Schönheim bound gives a floor:

$$C(v,k,t) \;\ge\; \left\lceil \frac{v}{k} \left\lceil \frac{v-1}{k-1} \cdots \left\lceil \frac{v-t+1}{k-t+1} \right\rceil \cdots \right\rceil \right\rceil$$

It is often not tight, which is precisely why the search continues. The smaller wheels bundled on this page were built here by greedy set cover — an approximation that verifies but makes no claim to minimality, and says so.

The invariance, which is the important part

Here is the result that keeps a wheel honest.

Take any $n$ distinct tickets. Against a uniformly random draw, every $k$-subset has the same distribution of matches — the game has no idea which numbers are on which slip. So by linearity of expectation, the expected total return is

$$\mathbb{E}\left[\sum_{i=1}^{n} R(T_i)\right] = \sum_{i=1}^{n} \mathbb{E}[R(T_i)] = n \cdot \mathbb{E}[R]$$

regardless of how the tickets are arranged. Linearity does not care about dependence, and the arrangement is pure dependence.

So a covering design cannot create a penny of expected value. What it does is move probability between outcomes: it raises the floor under the conditional and pays for it out of the tail, leaving the mean untouched. The two histograms on this page have the same average and different shapes, and that is the exact and complete description of what a wheel buys you.

A guaranteed prize is not a guaranteed profit. The guarantee is real; it is simply worth less than it costs.

Manchester, 2023

Two mathematicians at the University of Manchester decided to settle a question ordinary lottery players have been asking each other forever, usually in a pub, usually without expecting an answer: how many tickets would you need to buy to be certain of winning something?

David Cushing and David Stewart answered it. Twenty-seven. Not probably, not on average — certainly, for every one of the 45,057,474 draws a 6-from-59 game can produce. They also proved that twenty-six will not do, using an exhaustive constraint search that had to rule out every arrangement of twenty-six tickets in existence.

Then they did the thing that makes them this product's spirit animal. They bought the tickets.

On 1 July 2023 they played their own twenty-seven lines. The theorem held exactly as proven: they got three 2-matches, each paying a free Lucky Dip. The three free tickets then lost. Twenty-seven tickets at £2 is £54, and £54 is what they were down.

They published that too.

Why that is the right way round

It would have been easy to publish the theorem and stop. The result is genuinely lovely — three Fano planes and two triangles, a construction you can hold in your head — and the arithmetic afterwards adds nothing to it mathematically.

But the arithmetic is the part a reader needs. "Guaranteed win" is a phrase that has sold an enormous amount of nonsense, and a paper proving a guaranteed win is exactly the sort of thing that gets excerpted into an advertisement. By buying the tickets and reporting the loss in the same breath as the proof, they made the result very difficult to misquote.

That is a kind of rigour that has nothing to do with mathematics, and this playground has been trying to practise it for seventeen features.

The quiet archive

Behind the world's most disreputable storefront there is a real research field, and it has an address.

The La Jolla Covering Repository, maintained by Dan Gordon for decades, is where the best-known covering designs live. It is a plain, unglamorous, enormously useful archive — mathematics as public infrastructure — and it has been quietly supplying the raw material for commercial "wheeling systems" for about as long as it has existed. The systems are sold. The repository is free.

The lineage runs deeper still: Füredi, Székely and Zubor's bounds on the lotto problem in 1996; Li and van Rees's tables of lotto designs in 2002; Colbourn's surveys of covering designs. A genuine, active, respectable field, forever being mistaken for its own shadow.

Gail Howard, given her due

The largest commercial lottery-system empire ever built was half real.

Gail Howard's wheels were actual covering designs. That part was not a con at all — the guarantees were true, stated as conditionals, and the mathematics behind them was sound. Anyone who bought a wheel got a working wheel.

What was wrapped around them was different: the balanced-game claims, the hot and cold numbers, the advice about sums and skips and patterns — everything this playground spent sixteen features retiring. The wheels were the pocket the theorems lived in, and the theorems lent their credibility to everything else in the coat.

That is this playground's most durable lesson, and it belongs here at the end: the best-selling systems carry true theorems in their pockets. The presence of real mathematics somewhere in a product tells you nothing about the claim being made in the headline. You have to check which part is which.

The Workshop is the wheel without the wrapper. Same designs, same guarantees, same conditionals — and nothing at all about which numbers to choose, because the covering design never cared and neither should you. If you would rather not take our word for that, the pool selector above will take any twenty-seven numbers you like, including your birthday. The guarantee is identical.

The last word

Seventeen rooms ago, we promised you patterns. We showed you every one the mathematics allows: the shapes of counting, the faces of chance, the fingerprints of fraud, the geography of belief, the price of the crowd — and one small true guarantee that costs more than it pays. The balls never knew any of it. That was the pattern.