In a particular year, the month of January had exactly 4 Thursdays and 4 Sundays. On which day of the week did January 1st occur in the year?

To determine the day of the week on which January 1st fell in a year where there are exactly 4 Thursdays and 4 Sundays in January, we need to analyze the situation systematically.

1. ### Break Down the Problem

• January has 31 days.
• For there to be exactly 4 Thursdays and 4 Sundays, January can have at most 4 occurrences of these days.

2. ### Relevant Concepts

• If a month has 31 days, then it consists of 4 complete weeks (28 days), and the remaining 3 days can include any combination of the days in a week.
• Thus, the maximum occurrences for any day in January with 31 days is 5, which means if January has 4 Thursdays and 4 Sundays, the first day has to influence which specific days appear 5 times.

3. ### Analysis and Detail

1. Since January has 31 days, if January 1st is a certain weekday, the days of January can be laid out as follows:

   • Day 1: January 1
   • Day 2: January 2
   • Day 3: January 3
   • ...
   • Day 31: January 31

2. If January 1st is a Sunday:

   • Sundays: 1, 8, 15, 22, 29 (5 Sundays)

3. If January 1st is a Monday:

   • Sundays: 7, 14, 21, 28 (4 Sundays)

4. If January 1st is a Tuesday:

   • Sundays: 6, 13, 20, 27 (4 Sundays)

5. If January 1st is a Wednesday:

   • Sundays: 5, 12, 19, 26 (4 Sundays)

6. If January 1st is a Thursday:

   • Sundays: 4, 11, 18, 25 (4 Sundays)

7. If January 1st is a Friday:

   • Sundays: 3, 10, 17, 24, 31 (5 Sundays)

8. If January 1st is a Saturday:

   • Sundays: 2, 9, 16, 23, 30 (5 Sundays)

From the analysis, we see that:

• If January 1st is a Monday, Tuesday, or Wednesday, we have 4 Sundays.
• If January 1st is a Thursday, we also have 4 Sundays.

4. ### Verify and Summarize

The only scenario that fits the condition of having 4 Thursdays without exceeding 4 Sundays occurs when January 1st is either a Monday, Tuesday, or Wednesday.

Final Answer

Thus, January 1st must have occurred on either Monday, Tuesday, or Wednesday.