Mr 'X' has three children. The birthday of the first child falls on the 5th Monday of April, that of the second one falls on the 5th Thursday of November. On which day is the birthday of his third child, which falls on 20th December?

To solve this, we need to determine the specific day of the week for a reference date in the given year, based on the information provided.

1. Analyze the first child's birthday: "5th Monday of April". April has 30 days. For a month with 30 days to have a 5th occurrence of a weekday, that weekday must fall on the 1st or 2nd of the month.

   • If April 1st is Monday, then the 5th Monday is April 29th.
   • If April 1st is Sunday (making April 2nd Monday), then the 5th Monday is April 30th.
   • If April 1st is any other day, there won't be a 5th Monday in April. So, April 1st must be a Monday or a Sunday.

2. Analyze the second child's birthday: "5th Thursday of November". November has 30 days. Similar to April:

   • If November 1st is Thursday, then the 5th Thursday is November 29th.
   • If November 1st is Wednesday (making November 2nd Thursday), then the 5th Thursday is November 30th.
   • If November 1st is any other day, there won't be a 5th Thursday in November. So, November 1st must be a Thursday or a Wednesday.

3. Find the consistent calendar: We need to find a scenario where both conditions hold true for the same year. Let's calculate the number of "odd days" (remainder when divided by 7) between April 1st and November 1st.

   • April: 30 days (2 odd days)
   • May: 31 days (3 odd days)
   • June: 30 days (2 odd days)
   • July: 31 days (3 odd days)
   • August: 31 days (3 odd days)
   • September: 30 days (2 odd days)
   • October: 31 days (3 odd days) Total odd days = 2+3+2+3+3+2+3 = 18 odd days. 18 mod 7 = 4 odd days. This means November 1st will be 4 days after April 1st.

   Let's test the possibilities for April 1st:

   • If April 1st is Monday: Then November 1st would be Monday + 4 days = Friday. If November 1st is Friday, there are only 4 Thursdays in November (7th, 14th, 21st, 28th). This contradicts the condition of having a 5th Thursday. So, April 1st cannot be Monday.
   • If April 1st is Sunday: Then November 1st would be Sunday + 4 days = Thursday. If November 1st is Thursday, then the 5th Thursday is November 29th (1st, 8th, 15th, 22nd, 29th). This satisfies the condition. Therefore, in this year, April 1st is Sunday, and November 1st is Thursday.

4. Determine the day for December 20th: We know November 1st is Thursday. We need to find the day for December 20th. Number of days from November 1st to December 20th:

   • Remaining days in November (from Nov 1st): 30 - 1 = 29 days.
   • Days in December: 20 days. Total days = 29 + 20 = 49 days. 49 mod 7 = 0 odd days. This means December 20th falls on the same day of the week as November 1st.

   Since November 1st is Thursday, December 20th is also THURSDAY.

The final answer is B) THURSDAY