The dates of three Sundays of a month were even numbers. What day of the week was the $20$th of the month?
1. **Understanding the problem**: We are given that three Sundays in a month are even-numbered dates. We need to determine the day of the week for the 20th of that month.

2. **Analyzing the given information**: 
   - Any two consecutive Sundays differ by 7 days.
   - If three Sundays are even-numbered, they must be spaced by 14 days (i.e., 2 weeks apart) to maintain the even parity.

3. **Determining the possible Sundays**:
   - Let the three even-numbered Sundays be \(2k, 2k+14, 2k+28\) where \(k\) is an integer.
   - Since the maximum number of days in a month is 31, the highest possible even-numbered Sunday is 30.

4. **Finding the possible values for \(k\)**:
   - The smallest even number is 2, so \(2k = 2\).
   - The largest even number that can be a Sunday is 30, so \(2k + 28 \leq 30\).
   - Solving for \(k\), we get \(2k \leq 2\), which implies \(k = 1\).

5. **Identifying the Sundays**:
   - If \(k = 1\), the Sundays are \(2, 16, 30\).

6. **Determining the day of the week for the 20th**:
   - If the 2nd is a Sunday, then the days of the week for the month are:
     - 2nd: Sunday
     - 3rd: Monday
     - 4th: Tuesday
     - 5th: Wednesday
     - 6th: Thursday
     - 7th: Friday
     - 8th: Saturday
     - 9th: Sunday
     - 10th: Monday
     - 11th: Tuesday
     - 12th: Wednesday
     - 13th: Thursday
     - 14th: Friday
     - 15th: Saturday
     - 16th: Sunday
     - 17th: Monday
     - 18th: Tuesday
     - 19th: Wednesday
     - 20th: Thursday

Therefore, the 20th of the month is a Thursday.

The final answer is \(\boxed{\text{Thursday}}\).