Someone wants to group all possible lottery draw results based on what the second largest number in them is. Which group will contain the most draws?
The second number naturally refers to the second term in the conventional - ascending - list of drawn numbers 11

a) Let the second number in the draw be $M$. According to this, one number is drawn from the numbers smaller than $M$ - the number of which is $M-1$ - and three from the $90-M$ numbers larger than it. Thus, the smaller number can be chosen in $M-1$ ways, the first of the larger numbers in $90-M$ ways, the second from the remaining in $89-M$ ways, and the third in $88-M$ ways. The product $f(M)=(M-1) \cdot(90-M)(89-M)(88-M)$ - taking into account somewhat the temporal order of the draw - gives the number of ways a draw can unfold that meets our criteria, i.e., with the constraint that $M$ was drawn first in time, and the second was the one smaller than it. It is easy to show that $f(M)$ is six times larger than the number of conventional listings of the appropriate draws, but this does not affect the search for the $M$ that gives the largest $f(M)$, because we are considering the same multiple (six times) of the number in question for every $M$.

Let's consider the change in the value of $f$ if we replace $M$ with the number one less:

\[
\begin{aligned}
f(M)-f(M-1) & =f(M)-(M-2)(91-M)(90-M)(89-M)= \\
= & (90-M)(89-M)[(M-1)(88-M)-(M-2)(91-M)]= \\
= & (90-M)(89-M)(94-4 M)
\end{aligned}
\]

The first two factors are positive in all cases - since the possible values of $M$ are 2, 87, and the intervening lottery numbers - and the third factor is negative when $M \geq 23.5$. Therefore, if $M \leq 23$, then $f(M-1) < f(M)$, and if $M > 23$, then $f(M-1) > f(M)$, meaning the value of $f(M)$ increases up to $M=23$ and then decreases, reaching its maximum value at $M=23$.

Béla Zöldy (Budapest, I. István g. II. o. t.)[^0]


[^0]:    ${ }^{1} \mathrm{If}$ we examine the question for the "second from the top" lottery number, then the number complementary to 91 of the value obtained with the above interpretation gives the solution, because if we subtract every number of a draw from 91, we get another lottery draw - and in this case, the number that replaces the second term from the top of the original draw, listed in descending order, is the second from the bottom. Thus, the number $91-23=68$ appears most frequently as the second lottery number from the top.