6.1 Let's say that number A hides number B if you can erase several digits from A to get B (for example, the number 123 hides the numbers 1, 2, 3, 12, 13, and 23). Find the smallest natural number that hides the numbers 2021, 2120, 1220, and 1202.
# Answer: 1201201

Solution. Notice that the number contains at least two twos and one zero. If there are exactly two twos, then the zero must stand both between them and after them, but then there must be at least two zeros. Therefore, only on twos and zeros, we need 4 digits (either two twos and two zeros, or three twos and one zero).

If there is one one, then before it should be $2,0,2$, and after it - two more twos and a zero. At the same time, the zero stands both between the twos and after them (1220 and 1202), so we need one more two or one more zero, making it 8 digits in total.

Suppose there are two ones. If there are exactly two twos, then we need a one after these twos (2021), between them (2120), and before them (1220), which leads to a contradiction. Therefore, there must be at least three twos. If there are 6 digits in total, then the zero is the only one. Then it stands after two twos and before one. From the number 1220, we get that one of the ones stands before the first two twos and the zero, and from the number 2021, we get that the other one stands after the last two. Then we definitely get the arrangement - 122021, but alas, from it, we cannot get the number 2120. Therefore, one zero is not enough, and we already have 7 digits.

Suppose there are three ones. We need 4 more digits for twos and zeros, so we have 7 digits in total.

Suppose there are 7 digits. The first digit is not zero, so it must be 1. If the next digit is 0, then it is useless (in our numbers, 0 appears only after twos). Two ones in a row or two zeros make no sense, as in each of our numbers, 0 or 1 appear only once. Therefore, the second most significant digit is 2, followed by 0, and then 1. 1201... We have three more digits, one of which is definitely a two. From the number 1220, we understand that somewhere after it stands a zero, and from the number 2021, we understand that after it stands another one. The smallest such variant is 201, and it fits.