## Task A-3.3.

A number consisting only of the digits 2 and 3 is called a happy number. Thus, happy numbers are in sequence $2,3,22,23, \ldots$. Determine the 2050th happy number.
## Solution.

There are two one-digit happy numbers (2 and 3) and four two-digit (22, 23, 32, 33). Similarly, we conclude that there are $2^n$ happy $n$-digit numbers (since each of the $n$ digits can be chosen in 2 ways) for $n=4,5, \ldots, 10$.

The total number of happy numbers with at most 10 digits is

$$
2+4+8+\cdots+1024=2046
$$

Thus, the 2047th happy number is an 11-digit number whose all digits are 2 (i.e., 22222222222), while the 2050th happy number is 22222222233.

2 points