Problem 4.7. Schoolchildren Alexander, Boris, Sergey, Daria, and Elena attended choir classes from Monday to Friday. It is known that:

- in each of the five days, exactly three of the schoolchildren were present, and exactly two were absent
- no one was absent for two consecutive days and no one was present for three consecutive days
- Elena missed two more days than Boris;
- there was only one day when Alexander and Sergey were both present;
- on Monday, Sergey was at the class.

Who was present at the class on Friday?
Answer: Alexander, Boris, Daria.

Solution. Since no one was present for three consecutive days, the maximum number of days one can attend is 4. Four classes a week are possible only if Wednesday is skipped. Since it is not allowed to skip two days in a row, the minimum number of days one can attend is 2. Two classes a week are possible only if they are on Tuesday and Thursday. Elena missed two more days than Boris. Therefore, Elena attended exactly two days (these days must be Tuesday and Thursday), and Boris attended exactly four days (these days must be Monday, Tuesday, Thursday, and Friday). Let's enter this information into the table, and also indicate that Sergei was present on Monday.

|  | Mon | Tue | Wed | Thu | Fri |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Alexander |  |  |  |  |  |
| Boris | + | + | - | + | + |
| Sergei | + |  |  |  |  |
| Daria |  |  |  |  |  |
| Elena | - | + | - | + | - |

There are already two absentees on Wednesday: Boris and Elena. Therefore, everyone else was present on Wednesday.

|  | Mon | Tue | Wed | Thu | Fri |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Alexander |  |  | + |  |  |
| Boris | + | + | - | + | + |
| Sergei | + |  | + |  |  |
| Daria |  |  | + |  |  |
| Elena | - | + | - | + | - |

There is only one day when Alexander and Sergei were present at the same time - this is Wednesday. Since Sergei was present on Monday, Alexander was not present on Monday. Then he was present on Tuesday (because you cannot skip two days in a row) and was not present on Thursday (because no one was present for three consecutive days).

|  | Mon | Tue | Wed | Thu | Fri |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Alexander | - | + | + | - |  |
| Boris | + | + | - | + | + |
| Sergei | + | - | + |  |  |
| Daria |  |  | + |  |  |
| Elena | - | + | - | + | - |

Daria had to come on Monday to have three present. Moreover, on Tuesday there are already three present, so Sergei and Daria were not present on Tuesday.

|  | Mon | Tue | Wed | Thu | Fri |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Alexander | - | + | + | - |  |
| Boris | + | + | - | + | + |
| Sergei | + | - | + |  |  |
| Daria | + | - | + |  |  |
| Elena | - | + | - | + | - |

Alexander was definitely present on Friday (you cannot be absent for two days in a row). Then Sergei was not present on Friday (there is already a common day with Alexander) and was present on Thursday (you cannot skip two days in a row).

|  | Mon | Tue | Wed | Thu | Fri |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Alexander | - | + | + | - | + |
| Boris | + | + | - | + | + |
| Sergei | + | - | + | + | - |
| Daria | + | - | + |  |  |
| Elena | - | + | - | + | - |

On Thursday, there are three attendances, so Daria was not present on Thursday and was present on Friday (you cannot skip two days in a row).

|  | Mon | Tue | Wed | Thu | Fri |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Alexander | - | + | + | - | + |
| Boris | + | + | - | + | + |
| Sergei | + | - | + | + | - |
| Daria | + | - | + | - | + |
| Elena | - | + | - | + | - |

Thus, on Friday, the attendees were Alexander, Boris, and Daria.