9.7. On the board, 99 numbers were written, none of which are equal. In a notebook, $\frac{99 \cdot 98}{2}$ numbers were written—all differences between two numbers from the board (each time subtracting the smaller number from the larger one). It turned out that the number 1 was written exactly 85 times in the notebook. Let $d$ be the largest number written in the notebook. Find the smallest possible value of $d$.

(L. Samoilov)
Answer. $d=7$.

Solution. We will prove that $d \geqslant 7$. All numbers on the board can be divided into chains of numbers of the form $a, a+1, a+2, \ldots, a+t$ such that numbers from different chains do not differ by exactly 1. Such a partition is not difficult to construct by connecting any two numbers that differ by 1 with a segment and considering the resulting broken lines.

Let there be $k$ chains, in which there are $n_{1}, n_{2}, \ldots, n_{k}$ numbers respectively (some chains may consist of a single number). In a chain of $n_{i}$ numbers, there are exactly $n_{i}-1$ pairs of numbers that differ by 1. Therefore, the total number of ones in the notebook is

\[
\left(n_{1}-1\right)+\left(n_{2}-1\right)+\ldots+\left(n_{k}-1\right)=\left(n_{1}+n_{2}+\ldots+n_{k}\right)-k=99-k,
\]
from which $k=99-85=14$. Thus, in one of the chains, there are at least 99/14 numbers, that is, at least 8 numbers. The difference between the largest and smallest numbers in such a chain is at least $8-1=7$.

It remains to provide an example where $d=7$. Such an example is given, for example, by the numbers

\[
0=\frac{0}{14}, \frac{1}{14}, \frac{2}{14}, \ldots, \frac{98}{14}=7
\]

Indeed, in this example, $d=7$, and exactly the first 85 of these numbers in the set have a number that is one greater.

Remark. The provided example is not the only one. All possible optimal examples are structured such that there is exactly one chain of 8 numbers (from $a$ to $a+7$), and 13 chains, each of 7 numbers; all numbers of these other chains must be located between $a$ and $a+7$.

Comment. Only the answer - 0 points.

Providing only an example with $d=7$ that satisfies the conditions of the problem - 2 points.

Providing only the estimate showing that $d \geqslant 7$ - 5 points.

The statement that all numbers can be divided into chains, in each of which the numbers differ by 1 (and the differences between numbers from different chains are not equal to 1) is accepted without proof. The statement that in a chain of $n_{i}$ numbers there are no more than $n_{i}-1$ unit differences is also accepted without proof.