“Surely 23 and 24 won’t both come up.”

The trick is to squeeze the gaps out

Counting draws with neighbours directly is awkward. Counting the ones without them is easy, and then you subtract.

Take any selection of 5 numbers with no two side by side. Now squeeze one space out from between each chosen number: the first stays put, the second drops by one, the third by two, and so on. Because no two were adjacent to begin with, nothing collides — and what you are left with is an ordinary selection of 5 numbers from a smaller pool of $N - k + 1$.

The map runs both ways, so the two sets are exactly the same size:

$$\#\{\text{no neighbours}\} = \binom{N-k+1}{k}$$

Everything else follows:

$$P(\text{at least one adjacent pair}) = 1 - \frac{\binom{N-k+1}{k}}{\binom{N}{k}}$$

For Euro Millions Plus that is 35.3%. For the classic 6/49 game it comes to 49.52% — a coin flip, near enough that most people would take either side of the bet and be wrong to feel confident.

The whole shape, not just the headline

The same gap argument gives the exact number of tickets with exactly $j$ adjacent pairs. Choose which $j$ of the $k-1$ internal boundaries are fused, which leaves $k-j$ runs to place, and place them by the same squeeze:

$$\#\{\text{exactly } j\} = \binom{k-1}{j}\binom{N-k+1}{k-j}$$

Add those up over every $j$ and Vandermonde's identity hands back $\binom{N}{k}$ — every possible ticket, accounted for once. That identity is the cheapest available proof the formula is right, and the test suite asserts it for every format this site holds.

More than half a neighbour per draw

The average number of adjacent pairs is simpler still. There are $N-1$ places where a neighbouring pair could sit, each is fully drawn with probability $\binom{N-2}{k-2}/\binom{N}{k}$, and by linearity of expectation the whole thing collapses to:

$$E[\text{adjacent pairs}] = \frac{k(k-1)}{N}$$

Here that is 0.400. For 6/49 it is $30/49 \approx 0.612$ — on average, rather more than half a neighbouring pair in every single draw.

Why the intuition fails

Consecutive numbers look less random, and that feeling is doing all the work.

But randomness is a property of the process, not of how the outcome looks afterwards. The machine does not know that 23 and 24 are written next to each other on a number line; it has no concept of adjacency at all. Every specific spread-out ticket you might prefer is exactly as likely as 1-2-3-4-5-6, and there are simply a great many more spread-out tickets — which is the only reason spread-out draws are common.

Your surprise is a fact about you. It is not evidence about the machine.

Mathematicians took the question seriously

In 2005 Konstantinos Drakakis posted a paper to arXiv with the flat title Distances between the winning numbers in Lottery. It works through the combinatorics of gaps between drawn numbers — the same squeeze argument in the maths panel, done properly and in general.

That such a paper exists is itself the story. The question "do lottery numbers come up next to each other?" sounds like idle bar-room speculation, and the honest answer turned out to be so far from everyone's gut that it was worth writing down carefully.

Half a century of German machines agreed

Formulas are one thing; actual balls in actual drums are another.

The German Lotto 6/49 ran from 1955 to 2011 with 5,026 draws on record. Of those, 2,557 contained consecutive numbers — 50.9%. The closed form says 49.52%.

Fifty-six years of physical machinery, in a country with a national appetite for record-keeping, landing within one and a half percentage points of a formula derived from nothing but counting. The wall above does the same thing for this lottery, and it lands in the same place.

The birthday-paradox family

If this feels like a trick you have met before, it is. The birthday paradox — 23 people in a room and it is more likely than not that two share a birthday — runs on the same engine.

In both cases the mistake is counting the wrong thing. You instinctively ask "how likely is a specific collision?" and get a small number. The question that matters is "how likely is any collision?", and there are far more chances for one than there feel like. Six numbers offer five potential neighbour positions; twenty-three people offer 253 possible pairs. Small probabilities compound faster than intuition tracks.

Whole draws repeating is the same family, taken further — Bulgaria managed it in 2009.

The part that might actually matter

Here is the twist worth keeping.

Because people avoid tickets that look unrandom, combinations containing neighbours are probably underplayed. That does not make them likelier to win — nothing does — but if such a ticket wins, it is shared with fewer people. That is the only real edge in a lottery, and it has nothing to do with which numbers come out of the machine.

Popularity is subtle, though. In April 2008 an Ontario draw came up 40-41-42-43-44-45 and 239 people held that ticket, having chosen it precisely because it looked impossible. Neat sequences at round positions are overplayed. Scattered neighbours in the middle of the range are not.

What you now know

Most lottery players will go their whole lives believing that side-by-side numbers are a fluke. You have just watched half of a real lottery's history contain them.

Your surprise is not evidence.