Finding the Day of the Week for June 15, 1992

The question asks for the day of the week Sonu was born, specifically June 15, 1992. To solve this type of problem, we can use the concept of 'odd days'. Odd days are the number of days left after dividing the total number of days by 7 (since a week has 7 days).

Understanding Odd Days

A normal year has 365 days. When divided by 7, \(365 \div 7 = 52\) weeks and 1 day remainder. So, a normal year has 1 odd day.

A leap year has 366 days. When divided by 7, \(366 \div 7 = 52\) weeks and 2 days remainder. So, a leap year has 2 odd days.

Centuries also have a specific number of odd days:

The pattern of odd days repeats every 400 years. For example, 500 years have the same odd days as 100 years (5), 600 years as 200 years (3), and so on.

Calculating Odd Days up to June 15, 1992

We need to find the total number of odd days from a reference point (typically the beginning of the calendar, like Jan 1, 1 AD) up to June 15, 1992. We can break this down:

1. Odd days up to the end of the previous year (December 31, 1991).
2. Odd days in the target year (1992) up to the specific date (June 15).

1. Odd days up to December 31, 1991:

We consider the period of 1991 years. We can split this into centuries and remaining years:

• Odd days in 1900 years: \(1900 = 1600 + 300\) years.
  • Odd days in 1600 years (\(4 \times 400\)) = 0 odd days.
  • Odd days in 300 years = 1 odd day.
  • Total odd days in 1900 years = \(0 + 1 = 1\) odd day.
• Odd days in the remaining 91 years (1901 to 1991): We need to count the number of leap years and normal years in this period. A year is a leap year if it is divisible by 4, unless it is a century year not divisible by 400.
  • Leap years between 1901 and 1991 are 1904, 1908, ..., 1988. The number of leap years = \(\frac{1988 - 1904}{4} + 1 = \frac{84}{4} + 1 = 21 + 1 = 22\) leap years.
  • Normal years = Total years - Leap years = \(91 - 22 = 69\) normal years.
  • Odd days in 91 years = (Number of normal years \(\times\) 1) + (Number of leap years \(\times\) 2) = \((69 \times 1) + (22 \times 2) = 69 + 44 = 113\) days.
  • Odd days in 91 years = \(113 \div 7\) remainder = \(113 \pmod{7} = 1\) odd day.

Total odd days up to December 31, 1991 = Odd days in 1900 years + Odd days in 91 years = \(1 + 1 = 2\) odd days.

2. Odd days in the year 1992 up to June 15:

First, check if 1992 is a leap year. 1992 is divisible by 4, so it is a leap year. This means February has 29 days in 1992.

Now, count the days from January 1, 1992, up to June 15, 1992:

• January: 31 days \(\equiv 31 \pmod{7} \equiv 3\) odd days
• February: 29 days (leap year) \(\equiv 29 \pmod{7} \equiv 1\) odd day
• March: 31 days \(\equiv 31 \pmod{7} \equiv 3\) odd days
• April: 30 days \(\equiv 30 \pmod{7} \equiv 2\) odd days
• May: 31 days \(\equiv 31 \pmod{7} \equiv 3\) odd days
• June: 15 days \(\equiv 15 \pmod{7} \equiv 1\) odd day

Total odd days in 1992 up to June 15 = \(3 + 1 + 3 + 2 + 3 + 1 = 13\) odd days.

When we find the remainder when 13 is divided by 7: \(13 \pmod{7} = 6\) odd days.

Total Odd Days and Day of the Week

Total odd days up to June 15, 1992 = Odd days up to Dec 31, 1991 + Odd days in 1992 up to June 15

Total odd days = \(2 + 6 = 8\) odd days.

The final number of odd days is \(8 \pmod{7} = 1\).

Now we map the total odd days to the day of the week using a standard convention:

Since the total number of odd days is 1, the day of the week for June 15, 1992, is Monday.

Conclusion

Based on the calculation of odd days, Sonu was born on a Monday.

The final answer is \(\boxed{Monday}\).

Revision Table: Calendar Odd Days

Additional Information on Calendar Calculations

Calendar problems often involve calculating the day of the week for a given date. The odd days method is a common technique used for this.

Key concepts in calendar calculations include:

• Normal Year: A year with 365 days. It has 1 odd day.
• Leap Year: A year with 366 days, where February has 29 days. It has 2 odd days. A year is a leap year if it is divisible by 4, except for century years which must be divisible by 400. For example, 1900 was not a leap year, but 2000 was. 1992 was divisible by 4, so it was a leap year.
• Odd Days in Months: The number of odd days in a month depends on the number of days in that month (30 or 31, or 28/29 for February). \(30 \pmod{7} = 2\) odd days, \(31 \pmod{7} = 3\) odd days, \(28 \pmod{7} = 0\) odd days, \(29 \pmod{7} = 1\) odd day.
• Reference Day: The mapping of odd days (0=Sunday, 1=Monday, etc.) assumes a specific reference point. While the starting point of 1 AD is theoretical, the relative calculations based on odd days hold true for finding the day of the week for a given date.

Practicing these calculations helps in mastering calendar-based reasoning questions frequently found in exams.