4. Stem-and-leaf plot. (From 6th grade, 2 points). To represent whole numbers or decimal fractions, a special type of diagram called a "stem-and-leaf plot" is often used. Such diagrams are convenient for representing people's ages. Suppose that in the studied group, there are 5 people aged 19, 34, 37, 42, and 48. For this group, the diagram will look as shown in Fig. 2. The left column is the "stem," and to the right of it are the "leaves."

When studying a certain group of patients, on December 1, the doctor created a diagram of their ages (Fig. 3a). On Fig. 3b, a new diagram of their ages is shown, which was also created on December 1, several years later. Over these years, the composition of the group remained the same - all those who were there remained, and no one new joined the group. However, the numbers on the new diagram are not visible - they are replaced by asterisks. Determine how many years have passed and restore the diagram.

$$
\begin{array}{l|llllll}
0 & & & & & & \\
1 & 0 & 0 & 1 & 2 & 2 & 3 \\
2 & 1 & 5 & 6 & & & \\
3 & 0 & 2 & 4 & 6 & & \\
4 & 1 & 6 & & & &
\end{array}
$$

Fig. 3 a)

![](https://cdn.mathpix.com/cropped/2024_05_06_aa973e43eb29b804d001g-01.jpg?height=383&width=420&top_left_y=1439&top_left_x=1161)

Fig. 3 b)
Solution. The digits from 0 to 5, representing decades of years, can be placed immediately (Fig. 4a). It is clear that less than 10 years have passed, otherwise there would be no digits in line "1".

If 7 or more years had passed, then the person who is 13 years old would have moved to line "2", and there would be fewer than six values left in line "1". But this did not happen, so less than 7 years have passed.

No one from line "1" moved to line "2", but two people, aged 25 and 26, moved from line "2" to line "3". However, the number of values in line "3" did not change, which means that those who were 34 and 36 years old moved to line "4". Therefore, at least 6 years have passed.

Thus, exactly 6 years have passed, and the diagram shown in Fig. 4b) was obtained. [^0]

![](https://cdn.mathpix.com/cropped/2024_05_06_aa973e43eb29b804d001g-02.jpg?height=366&width=422&top_left_y=268&top_left_x=677)

Fig. 4 a)

![](https://cdn.mathpix.com/cropped/2024_05_06_aa973e43eb29b804d001g-02.jpg?height=366&width=351&top_left_y=268&top_left_x=1184)

Fig. 4 b)

Answer: 6 years, see Fig. 4 b).