Fourteen friends met at a party. One of them, Fredek, wanted to go to bed early. He said goodbye to 10 of his friends, forgot about the remaining 3 , and went to bed. After a while he returned to the party, said goodbye to 10 of his friends (not necessarily the same as before), and went to bed. Later Fredek came back a number of times, each time saying goodbye to exactly 10 of his friends, and then went back to bed. As soon as he had said goodbye to each of his friends at least once, he did not come back again. In the morning Fredek realised that he had said goodbye a different number of times to each of his thirteen friends! What is the smallest possible number of times that Fredek returned to the party?
Answer: Fredek returned at least 32 times.

Assume Fredek returned $k$ times, i.e. he was saying good-bye $k+1$ times to his friends. There exists a friend of Fredek, call him $X_{13}$, about whom Fredek forgot $k$ times in a row, starting from the very first time - otherwise Fredek would have come back less than $k$ times.

Consider the remaining friends of Fredek: $X_{1}, X_{2}, \ldots, X_{12}$. Assume that Fredek forgot $x_{j}$ times about each friend $X_{j}$. Since Fredek forgot a different number of times about each of his friends, so we can assume w.l.o.g. that $x_{j} \geqslant j-1$ for $j=1,2, \ldots, 12$. Since $X_{13}$ was forgotten by Fredek $k$ times, and since Fredek forgot about exactly three of his friends each time, we have

$$
\begin{aligned}
3(k+1) & =x_{1}+x_{2}+x_{3}+\ldots+x_{12}+k \geqslant \\
& \geqslant 0+1+2+3+\ldots+11+k= \\
& =66+k .
\end{aligned}
$$

Therefore $2 k \geqslant 63$, which gives $k \geqslant 32$.

It is possible that Fredek returned 32 times, i.e. he was saying good-bye 33 times to his friends. The following table shows this. The $i$-th column displays the three friends Fredek forgot while saying good-bye for the $i$-th time (i.e. before his $i$-th return). For simplicity we write $j$ in place of $X_{j}$.

$$
\begin{aligned}
& 13 \ldots .131313 \ldots .1313 \ldots .131313131313131311
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_04_17_8c3840558e5986866c27g-08.jpg?height=43&width=869&top_left_y=1310&top_left_x=211)

![](https://cdn.mathpix.com/cropped/2024_04_17_8c3840558e5986866c27g-08.jpg?height=80&width=871&top_left_y=1354&top_left_x=208)