For different numbers. Vasya pays one ruble each time to choose any three cards, and asks Borya to arrange them in increasing order of the numbers written on their backs. Can Vasya achieve the following goal by paying 2000 rubles: to arrange all 365 cards in increasing order of the numbers written on their backs on the table?
To prove that Vasya can achieve his goal, we first state a lemma.

Lemma: Suppose Vasya pays \( x \) rubles to arrange \( N-1 \) cards in the required order on the table, where \( N \leqslant 3^{k} \). Then he needs to pay no more than \( k \) rubles to place the remaining one card in the correct position, meaning he needs to pay no more than \( x + k \) rubles in total.

We will prove this lemma by induction. The case \( k=1 \) is obvious. Assume the lemma holds for all \( N \leqslant 3^{k-1} \), and we will prove it for \( N \leqslant 3^{k} \). Clearly, Vasya's first step should be to determine the position of the \( N \)-th card relative to the cards \( A \) and \( B \) placed at the \( \left[\frac{N}{3}\right] \) and \( \left[\frac{2N}{3}\right] \) positions, respectively. Since cards \( A \) and \( B \) divide the already arranged cards into three segments, Vasya can use 1 ruble to determine which segment the \( N \)-th card belongs to. Since each segment contains no more than \( 3^{k-1} - 1 \) cards, by the induction hypothesis, Vasya needs to pay no more than \( k-1 \) rubles to further determine the position of the \( N \)-th card within that segment. The lemma is proved.

Next, we use the lemma to prove that Vasya can achieve his goal by spending no more than 2000 rubles. To arrange the first three cards, he spends 1 ruble. To insert the 4th to 9th cards (a total of 6 cards), Vasya spends no more than 2 rubles per card. To insert the 10th to 27th cards, he spends no more than 3 rubles per card. To insert the 28th to 81st cards, he spends no more than 4 rubles per card. Next, for the 82nd to 243rd cards, he spends no more than 5 rubles per card, and for the 244th to 365th cards, he spends no more than 6 rubles per card. This means Vasya needs to spend
\[
\begin{aligned}
& 1 + 6 \times 2 + 18 \times 3 + 54 \times 4 + 162 \times 5 + 122 \times 6 \\
= & 1845 < 2000
\end{aligned}
\]
rubles in total to achieve his goal.