$\underline{\text { Customer }}$ O.H.

Detective Nero Wolfe is investigating a crime. The case involves 80 people, one of whom is the criminal, and another is a witness to the crime (but it is unknown who they are). Each day, the detective can invite one or several of these 80 people, and if the witness is among the invited but the criminal is not, the witness will reveal who the criminal is. Can the detective definitely solve the case in 12 days?
The first method. We will number the 80 suspects with four-digit ternary numbers $\overline{a b c d}$ from 0001 to 2222 (there are exactly $3^{4}-1=80$ such numbers).

On the $k$-th day ($k=1,2, \ldots, 12$), Nero Wolfe invites those suspects whose numbers satisfy the $k$-th of the equalities $a=0, a=1, a=2$,

$b=0, b=1, b=2, c=0, c=1, c=2, d=0, d=1, d=2$. Then, on one of these days, he will invite the witness, but not the criminal, since their numbers differ in at least one digit.

The second method. We will divide the suspects into 16 groups of 5 people each. We will number the groups with four-digit binary numbers $\overline{a b c d}$ from 0000 to 1111, and we will number the members of each group from 1 to 5.

On the $k$-th day ($k=1,2, \ldots, 8$), Nero Wolfe invites suspects from groups whose numbers satisfy the $k$-th of the equalities: $a=0, a=1$,

$b=0, b=1, c=0, c=1, d=0, d=1$. If the criminal and the witness are in different groups, the crime will be solved on one of these days.

Suppose the witness is in the same group as the criminal. Let $m$ and $n$ be the numbers of the witness and the criminal in this group, respectively. On the 9th day, Nero Wolfe invites suspects with numbers 1 and 2 from each group, on the 10th day - suspects with numbers 3 and 4, on the 11th day - suspects with numbers 1, 3, and 5, and on the 12th day - suspects with numbers 2, 4, and 5. Then, there will be one of these days when suspects with number $m$ are invited from each group, but suspects with number $n$ are not invited. This means that on this day, the crime will be solved.

The third method. We will show that Nero Wolfe can find the criminal even in 9 days. For this, we will assign each suspect a number - a nine-digit binary number, four digits of which are ones, and five are zeros (this is possible since there are $C_{9}^{4}=126$ such numbers).

On the $k$-th day ($k=1,2, \ldots, 9$), Nero Wolfe invites those suspects whose $k$-th digit of the number is 1. Since all numbers contain exactly 4 ones, there will be a number $m$ from 1 to 9 such that the witness has a 1 in the $m$-th place, and the criminal has a 0. This means that on the $m$-th day, the witness will be invited to Nero Wolfe without the criminal, and the crime will be solved.

## Answer

Yes.