## 57. How Many Rattles?

- My head is aching from the last problem, - complained the White Queen. - Let's go back to arithmetic problems. Are you familiar with Tralala and Trulala?
- Of course! - rejoiced Alice.
- Excellent! So, one day Tralala and Trulala made a bet.
- What did they bet on? - asked Alice.
- The bet arose over a scary raven. Tralala thought the raven would definitely come the next day, while Trulala thought the raven wouldn't come. So they decided to make a bet.
- And what did they bet?
- You must know that both brothers, Tralala and Trulala, collect rattle collections.
- I know, - answered Alice, - that Trulala had a rattle and claimed that Tralala broke it, but I didn't know that Tralala also collects rattles.
- They both collect rattles, - confirmed the Queen. - Each has several. So, they bet on one rattle.
- How funny! - laughed Alice. - And how many rattles does each have?
- That's exactly what you need to find out! - answered the Queen. - That's the problem. Tralala knows that if he loses the bet, he will have as many rattles as Trulala, and if he wins the bet, he will have twice as many rattles as Trulala.

How many rattles does Tralala have and how many does Trulala have?

Usually, this problem is solved using algebra, but Alice hadn't studied algebra in school yet. Fortunately, she had learned something from the Griffin and managed to solve the problem successfully.

So, how many rattles does Tralala have and how many does Trulala have?
57. How many rattles? If Tra-la-la loses the bet, he will have half of the total number of rattles (or, the same, as many rattles as Tru-li-la), so before the bet, Tra-la-la has one rattle more than Tru-li-la. If Tra-la-la wins the bet, he will have two more rattles than half of the total number of rattles. Moreover, after winning, Tra-la-la will have $2 / 3$ of the total number of rattles (or, the same, twice as many rattles as Tru-li-la), which is $1 / 6$ of the total number of rattles more than half of the total number of rattles ($1 / 2 - 1 / 3 = 1 / 6$). Therefore, the increment of $1 / 6$ over half of the total number of rattles is the same as the increment of 2 rattles over half of the total number of rattles. Thus, the total number of rattles is 12, so Tra-la-la has 7 rattles, and Tru-li-la has 5.

Verification. If Tra-la-la loses the bet, each of the brothers will have 6 rattles. If Tra-la-la wins the bet, he will have twice as many rattles as Tru-li-la—8 against 4.