How many 100-digit natural numbers are there, all of whose digits are even and the difference between any two adjacent digits is $2$?
Let's try to write down these numbers! First, let the first digit be 2 or 6. In both cases, there are 3 possibilities for the second and third digits: 202, 242, 246, and 686, 642, 646. Notice that the third digit is also either 2 or 6. The fourth and fifth digits can then be chosen in 3 ways, and so on. Thus, after every odd-numbered digit, we have 3 possibilities for the next two digits. This is a total of 49 times (after the 1st, 3rd, 5th, ..., 97th digit). The 99th digit will again be 2 or 6, so the 100th digit can be chosen in 2 ways. This results in \(3^{49} \cdot 2\) cases for both 2 and 6 as the starting digit. The 100-digit numbers thus written are all different, and our method leads to all 2- or 6-starting numbers that meet the problem's conditions.

If the number starts with 4 or 8, it can be continued as follows: 42, 46, 86. Here, the second digit is 2 or 6, so there are 3 possibilities for the third and fourth digits, the fourth digit is 2 or 6, and so on. Thus, after each even-numbered digit (2nd, 4th, ..., 98th), the next two digits can again be 3 ways, which is \(3^{49}\) cases for each of the three starting digits. Each case results in a different number, and the method gives all 100-digit numbers starting with 4 or 8, where consecutive digits are even and their difference is 2.

14So, 2-starting: \(3^{49} \cdot 2\) numbers,6-starting: \(3^{49} \cdot 2\) numbers,42-starting: \(3^{49}\) numbers,46-starting: \(3^{49}\) numbers,86-starting: \(3^{49}\) numbers,

in total, we have \(7 \cdot 3^{49}\) such numbers.

Based on the idea of Váradi Vajk (Győr, Czuczor G. Bencés Gimn., 9th grade)