# 6.1. Condition:

Petya thought of a natural number and wrote down the sums of each pair of its digits on the board. After that, he erased some of the sums, and the numbers $2,0,2,2$ remained on the board. What is the smallest number Petya could have thought of?
Answer: 2000

## Solution.

Since among the sums there is a 0, the number must contain at least two digits 0. If the number has only three digits, there will be three pairwise sums, while the condition states there are at least four. Therefore, the number must have at least four digits. A sum of 2 can be obtained either as $1+1$ or as $2+0$. If the number contains the digit 2, then it will have four digits, three of which are $-2,0,0$, which means the number cannot be less than 2000. At the same time, 2000 meets the condition. If 2 is obtained as $1+1$, then at least four digits $-1,1,0,0$ are required. With these, it is impossible to get pairwise sums of $2,0,2$, 2, which means there must be at least 5 digits, i.e., the number will definitely be greater than 2000.