Lottery players often complain that they didn't win again, especially | which is particularly frustrating | the numbers next to the ones they played were drawn. Let's examine the question in detail: were they really that close to the jackpot?

Assume that in the 5-number lottery, the winning numbers are: $7, 13, 28, 46, 75$. How many different ways can a lottery ticket be filled out such that at least four of the marked numbers are adjacent to the listed numbers? (Two numbers are adjacent if the absolute value of their difference is 1.)
The neighbors of $7, 13, 28, 46, 75$ are: 6, $8 ; 12,14 ; 27,29 ; 45,47 ; 74,76$; this is a total of 10 numbers, and apart from these, there are still 80 numbers left. The fact that at least four of the drawn numbers are among the neighboring numbers means that at most 1, but it could be that no drawn number appears among the other 80 numbers.

We distinguish two cases:

a) A number's both neighbors can appear, or for each number, at most only one neighbor can appear in the draw.

If both neighbors can appear, and there is no marked number among the other 80; this

$$
\binom{10}{5}=\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}=252 \text{ ways }
$$

is possible.

If there is one marked number among the 80, this is 80 cases, and the other 4 numbers are from the 10 neighbors; this is $\binom{10}{4}$ cases, so the total number of cases is;

$$
80\binom{10}{4}=16800
$$

Thus, in the first case, the number of possibilities is 17052.

b) In the second interpretation, if only one neighbor can appear for each number, and there is no marked number among the other 80, the number of possible cases is $2^{5}$. If there is a number among the 80, then the number of possible draws is $80 \cdot 5 \cdot 2^{4}$, which is a total of

$$
80 \cdot 5 \cdot 2^{4}+2^{5}=6432
$$

We can see that in both cases, there are quite a few ways to fill out the draw such that the numbers we bet on are neighbors of the winning numbers. (This is why it's not really worth getting too upset about such an outcome of the draw...)

Based on the work of Paál Judit, (Siófok, Perczel M. Gymn., IV. grade).

Remarks. In the case of "complainers" about the lottery draw, it is likely that the winning numbers did not appear in their draw, so the "remainder" is not 80, but only 75 numbers.

2. The problem would be slightly modified if the number 1 or 90 were among the drawn numbers, as these have only one neighbor.