A man starts work on Monday. He works for 8 consecutive days and then takes…

Concept: When an event repeats with a fixed period, the weekday it falls on advances by (period mod 7) each time it recurs. Here the man's schedule repeats every 9 days (8 working days plus 1 holiday), so consecutive holidays are 9 days apart, and after n complete cycles the n-th holiday falls on day 9n of the schedule.

Application:

1. The work-holiday cycle length is 8 + 1 = 9 days.

2. The 12th holiday falls on day 9 × 12 = 108 of the schedule, counting the starting Monday as day 1.

3. To find the weekday of day 108, divide by 7: 108 = 15 × 7 + 3, so day 108 is 2 days after a Monday, since (108 − 1) mod 7 = 2.

4. Counting 2 days after Monday — Monday, Tuesday, Wednesday — lands on Wednesday.

Cross-check: Each successive holiday shifts by 9 mod 7 = 2 weekdays from the previous one. Starting from the 1st holiday (day 9, a Tuesday), the sequence of holidays runs Tuesday, Thursday, Saturday, Monday, Wednesday, Friday, Sunday and then repeats every 7 holidays. Since 12 mod 7 = 5, the 12th holiday falls on the same weekday as the 5th holiday in this list, which is Wednesday — confirming the result independently.

Result: The man's 12th holiday falls on Wednesday.