5. The letters А, Б, К, М, П, У, Ш were encoded with sequences of zeros and ones (each with its own). Then, in the word ПАПАМАМАБАБУШКА, the letters were replaced with their codes. Could the length of the resulting sequence be shorter than 40 characters, if the sequence can be uniquely decoded?

保留源文本的换行和格式，所以翻译结果如下：

5. The letters А, Б, К, М, П, У, Ш were encoded with sequences of zeros and ones (each with its own). Then, in the word ПАПАМАМАБАБУШКА, the letters were replaced with their codes. Could the length of the resulting sequence be shorter than 40 characters, if the sequence can be uniquely decoded?
Answer: She could.

## Solution:

Here is an example of a code table:

| A | Б | К | М | П | У | Ш |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 0 | 110 | 1111 | 100 | 101 | 11100 | 11101 |

The word will look like this:

10101010100010001100110111001110111110 - a total of 38 characters. Decoding is unambiguous. To verify this, start from the left edge. The far left can only be the code 101, as no other code starts with the same combination of symbols. Next, similarly, as no code starts with a combination of symbols that matches any other code.

## Comment:

A uniform code (3 symbols per letter) gives 45 characters. Therefore, it is necessary to use codes of variable length for different letters. The more frequently a letter appears, the shorter its code should be. This consideration can help with verification. Be careful.

Participants can justify the unambiguousness of decoding not by sequentially examining the code sequence (from left to right), but by referring to a correct fact about such codes. The given code is an example of a prefix code - the code of any letter is not the beginning of the code of any other letter (Fano's condition), and for it, the theorem of unambiguous decoding is true. A reference to the corresponding theorem can be counted as justification (naturally, provided that it is correctly formulated).

## Instructions for Checking:

Score - either 0 if the answer is incorrect or the code table does not give the desired result, or 4 points if the codes are correct but there is no justification for the unambiguousness of decoding, or 7 points.