In a certain January, there are exactly 4 Thursdays and 4 Sundays. On which…

Concept. A month with 31 days equals 4 full weeks (28 days) plus 3 extra days. So every one of the 7 weekdays appears at least 4 times, and exactly the first 3 weekdays of the month appear a 5th time. Those three "5-time" weekdays are always three consecutive days beginning with the weekday of the 1st.

Application. We are told Thursday and Sunday each appear only 4 times. So neither Thursday nor Sunday may be one of the three consecutive 5-time days. We test where the block of three consecutive days can sit so that it skips both Thursday and Sunday:

1. If the 1st is Tuesday, Wednesday or Thursday, the three-day block contains Thursday, giving 5 Thursdays. Rejected.

2. If the 1st is Friday, Saturday or Sunday, the three-day block contains Sunday, giving 5 Sundays. Rejected.

3. If the 1st is Monday, the three 5-time days are Monday, Tuesday and Wednesday. This block contains neither Thursday nor Sunday, so both Thursday and Sunday appear exactly 4 times. Accepted.

Cross-check. Take the 1st as Monday. Then the Thursdays fall on the 4th, 11th, 18th and 25th — exactly 4 Thursdays (the next would be the 32nd, which does not exist). The Sundays fall on the 7th, 14th, 21st and 28th — exactly 4 Sundays (the next would be the 35th). Both counts equal 4, confirming the start day.

Result: 1st January is a Monday.