9.4. Vasya, a truant, skipped one lesson every Monday in September, two lessons every Tuesday, three on Wednesday, four on Thursday, and five on Friday. The exception was September 1: on this day, Vasya did not skip any lessons in honor of the start of the school year. It turned out that Vasya missed exactly 64 lessons for the entire month of September. What day of the week was September 1? Justify your answer. (All Saturdays and Sundays in September were holidays, and the other days were school days.)
Solution: Note that in any consecutive seven days, each day of the week occurs exactly once, so in any consecutive 7 days (starting from September 2), Vasya skips $1+2+3+4+5=15$ lessons. Therefore, from September 2 to September 29, he skips exactly $15 \cdot 4=60$ lessons, so on September 30, he skips $64-60=4$ lessons. This means that September 30 falls on a Thursday. Then the following dates in September also fall on a Thursday (subtracting 7 sequentially): $23, 16, 9, 2$. This means that September 1 fell on a Wednesday.

Answer: On Wednesday.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and justified answer | 7 points |
| Correct reasoning with an incorrect answer due to an incorrect number of days in September (31 instead of 30) | 6 points |
| Noted that in ANY consecutive 7 days, Vasya skips 15 lessons | 4 points |
| Shown that September 1 can fall on a Wednesday, but not shown that it cannot be another day | 2 points |
| Noted that September 1 and September 29 fall on the same day of the week; no further progress | 1 point |
| Correct answer without justification (with incorrect justification) | 0 points |