Understanding Day of the Week Calculation

This question asks us to find the day of the week on a past date, given the day on a future date. We are told that 16 January 2015 was a Friday, and we need to find the day of the week on 16 January 2010.

To solve this, we use the concept of "odd days". An odd day is the remainder when the total number of days between two dates is divided by 7. Since the days of the week repeat every 7 days, the number of odd days tells us how many days forward or backward the day of the week shifts.

• A normal year has 365 days, which is \(52 \times 7 + 1\) days. So, a normal year has 1 odd day.
• A leap year has 366 days, which is \(52 \times 7 + 2\) days. So, a leap year has 2 odd days.

Leap years are generally divisible by 4, except for century years which must be divisible by 400.

Calculating Odd Days Between 2010 and 2015

We need to find the number of odd days between 16 January 2010 and 16 January 2015. This period covers the years 2010, 2011, 2012, 2013, and 2014 completely from Jan 16 to Jan 15 of the next year, plus one day to reach Jan 16, 2015.

Alternatively, we can consider the number of years between the two dates and identify the leap years within this period. The period is from 16 January 2010 to 16 January 2015, which is exactly 5 years.

Let's list the years and check if they are leap years:

• 2010: Not divisible by 4. Normal year (1 odd day).
• 2011: Not divisible by 4. Normal year (1 odd day).
• 2012: Divisible by 4. Leap year (2 odd days). The leap day (Feb 29) falls within the period from 16 Jan 2010 to 16 Jan 2015.
• 2013: Not divisible by 4. Normal year (1 odd day).
• 2014: Not divisible by 4. Normal year (1 odd day).

We are moving from 16 Jan 2010 to 16 Jan 2015. The total duration is 5 years. Within this period, the year 2012 is a leap year, and its Feb 29 is included in the duration.

Number of normal years = 4 (2010, 2011, 2013, 2014)

Number of leap years = 1 (2012)

Total number of odd days = (Number of normal years \(\times\) Odd days in a normal year) + (Number of leap years \(\times\) Odd days in a leap year)

Total odd days = \((4 \times 1) + (1 \times 2) = 4 + 2 = 6\) odd days.

Finding the Day on 16 January 2010

We know the day on 16 January 2015 (Friday) and calculated the total odd days between 16 January 2010 and 16 January 2015 is 6. Since we are finding the day on an earlier date (2010) from a later date (2015), we need to move backward in the days of the week.

We need to go back 6 days from Friday.

• 1 day back from Friday is Thursday.
• 2 days back from Friday is Wednesday.
• 3 days back from Friday is Tuesday.
• 4 days back from Friday is Monday.
• 5 days back from Friday is Sunday.
• 6 days back from Friday is Saturday.

Alternatively, going back 6 days is the same as going forward \(7 - 6 = 1\) day.

1 day forward from Friday is Saturday.

Therefore, 16 January 2010 was a Saturday.

Summary of Steps

1. Identify the start and end dates: 16 January 2010 and 16 January 2015.
2. Determine the total number of years between the dates: 5 years.
3. Identify the leap years occurring between these dates (inclusive of the start date's year if Feb 29 falls within the period, exclusive of the end date's year if Feb 29 doesn't fall within the period, considering the exact dates). In this case, 2012 is the leap year, and its Feb 29 falls within Jan 16, 2010 to Jan 16, 2015.
4. Calculate the total odd days: (Number of normal years \(\times\) 1) + (Number of leap years \(\times\) 2). Total odd days = (4 \(\times\) 1) + (1 \(\times\) 2) = 6.
5. Adjust the day of the week. Since we are moving backward in time from 2015 to 2010, subtract the odd days from the day in 2015 (Friday).
6. Friday - 6 days = Saturday.

Calendar Odd Day Calculation Revision

Additional Calendar Information

Understanding leap years and odd days is crucial for solving calendar-based reasoning problems. A leap year occurs every 4 years to synchronize the calendar year with the solar year. The extra day is added in February.

• The Gregorian calendar rule for leap years: A year is a leap year if it is divisible by 4, unless it is divisible by 100 but not by 400.
• Example: 2000 was divisible by 400, so it was a leap year. 1900 was divisible by 100 but not by 400, so it was not a leap year.
• When calculating the day of the week for a date backward in time, you subtract the total number of odd days from the day of the week of the known date. If the result is negative, add 7 until it becomes positive.
• When calculating forward in time, you add the odd days. If the result is greater than 6 (assuming Sunday=0, Saturday=6), take the remainder when divided by 7.