A set of positive integers is called tyû-de-good if there are no two numbers among them whose difference is 2. How many tyû-de-good subsets does the set $\{1,2,3, \ldots, 10\}$ have?
Let's consider the even numbers $(2,4,6,8,10)$ and the odd numbers $(1,3,5,7,9)$ separately.

If we provide the number of tyú-de-jó subsets for both subsets (which are the same in any case), then the product of these two numbers will be the answer, since if the difference between two numbers is 2, then either both are even or both are odd (i.e., any even tyú-de-jó subset can be paired with any odd tyú-de-jó subset, and the tyú-de-jó property will still hold).

Let's just look at the odd numbers.

First, let the entire set be $\{1\}$. This set has 2 subsets, one containing 1 and the other not containing 1.

Now, let the entire set be $\{1,3\}$. Of the subsets of $\{1\}$, we can only add 3 to the one that does not contain 1. There is 1 such subset. But for this subset, it is also possible that we do not add 3. There is also 1 subset where 1 is included, to which we cannot add 3. Thus, there are $2(1+1$, i.e., the number of subsets of $\{1\}$) subsets to which we did not add 3, and 1 subset to which we did add 3 (where 1 is not included). Continuing this logic, we can see that the number of tyú-de-jó subsets increases according to the Fibonacci sequence when we add another odd number to the set. For example, for the next odd number, 5: if 5 is included in $\{1,3,5\}$, then 3 cannot be included, only the smaller ones, in this case, 1. If 5 is not included in the selected subset, then we get the 3 cases we obtained for 3. Thus, the number of tyú-de-jó subsets of $\{1,3,5,7,9\}$ is the third Fibonacci number after 3, which is 13.

Similarly, the set $\{2,4,6,8,10\}$ has 13 tyú-de-jó subsets. Therefore, the set $\{1,2,3, \ldots, 10\}$ has $13 \cdot 13 = 169$ tyú-de-jó subsets.

Based on the work of Sebestyén Pál Botond (Budapest, Baár-Madas Református Gimn., 8th grade)

Note: Some contestants also showed that if $a_{n}$ denotes the number of tyú-de-jó subsets of the set $\{1,2,3, \ldots, n\}$, then

$$
a_{n}=a_{n-2}+a_{n-3}+a_{n-4}
$$