Two siblings, starting from the age of 4, receive as many books on their birthdays as the number of the year they are in. How old are the children when the total number of books they have received on their birthdays is $100? $
Let's create the following table, which gives the possible age of one of the siblings and the total number of their books:

| How old | $0-3$ | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | $>15$ |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :--- |
| How many books | 0 | 4 | 9 | 15 | 22 | 30 | 39 | 49 | 60 | 72 | 85 | 99 | $>100$ |

The total number of books of the two siblings is the sum of any two numbers in the lower row. From the table, it is clear that this can only be 100 if one is 15 and the other is 85. Adding any other two numbers, or even taking the double of any one number (twins!), results in a value either greater or smaller than 100.

Therefore, the two children are 6 and 13 years old.

Based on the work of Ágnes Kovács (Kecskemét, Bányai Júlia Gymnasium, 8th grade)

Remarks. 1. We only gave 5 points if the solver indicated that this is the only solution.

2. Several people approached the problem by writing the number of books of an $n$-year-old child as the sum of the following arithmetic sequence:

$$
4+5+6+\cdots+n=\frac{n(n+1)}{2}-6
$$

They then tried to solve the equation with two unknowns:

$$
\frac{n(n+1)}{2}+\frac{m(m+1)}{2}-12=100
$$

($n, m$ are positive integers.) However, solving this is more complicated than the above table.