(20 points)] In a certain family, dad works on a "2 out of 2" schedule (2 days working, 2 days off), mom works on a "1 out of 2" schedule (1 day working, 2 days off), and the children attend school on a five-day workweek (Monday to Friday). On Saturday, September 2, mom visited grandma in the village, and the next day the whole family stayed at home. On which day will they have a common day off again?
Problem 864
Official solution
Answer: September 23 (Saturday).
Solution. According to the condition, September 2 and 3 were holidays for mom, so her working days later in September are the 4th, 7th, 10th, etc. On Sunday, September 3, it was a holiday for dad, but it is unknown whether he had a holiday the previous day. Let's consider 2 cases.
1) If September 2 was a holiday for dad, then his working days later in September are the 4th, 5th, 8th, 9th, 12th, 13th, etc. Let's mark in the calendar which Saturdays and Sundays the parents work:
| Date | 2 (Sat) | 3 (Sun) | 9 (Sat) | 10 (Sun) | 16 (Sat) | 17 (Sun) | 23 (Sat) | 24 (Sun) |
|---|---|---|---|---|---|---|---|---|
| Mom | P | P | ||||||
| Dad | P | P | P | P |
2) If dad worked on September 2, then his working days later in September are the 5th, 6th, 9th, 10th, 13th, 14th, etc. In this case, we get the following table:
| Date | 2 (Sat) | 3 (Sun) | 9 (Sat) | 10 (Sun) | 16 (Sat) | 17 (Sun) | 23 (Sat) | 24 (Sun) |
|---|---|---|---|---|---|---|---|---|
| Mom | P | P | ||||||
| Dad | P | P | P | P |
In both cases, the next common holiday in the family will be Saturday, September 23.