Understanding the Calendar Problem

This question asks us to find the day of the week for a past date (1st January 2016) given the day of the week for a future date (1st January 2017). To solve this, we need to understand how the day of the week shifts over a period of time, which depends on the total number of days in that period.

Key Concepts: Leap Years and Odd Days

The day of the week repeats every 7 days. The shift in the day of the week over a period depends on the remainder when the total number of days is divided by 7. This remainder is often called the number of 'odd days'.

• A normal year has 365 days. \(365 = 52 \times 7 + 1\). So, a normal year has 1 odd day. This means the day of the week shifts forward by 1 day from one year to the next (e.g., if Jan 1st 2023 was Sunday, Jan 1st 2024 would be Monday).
• A leap year has 366 days (due to the extra day in February). \(366 = 52 \times 7 + 2\). So, a leap year has 2 odd days. This means the day of the week shifts forward by 2 days from one year to the next.

We need to identify if the period between 1st January 2016 and 1st January 2017 includes a leap year and calculate the total number of days.

Identifying the Leap Year

A year is a leap year if it is divisible by 4, except for century years (years divisible by 100) which are leap years only if they are divisible by 400.

• The year 2016 is divisible by 4. \(2016 \div 4 = 504\). Therefore, 2016 was a leap year.
• The year 2017 is not divisible by 4. Therefore, 2017 was not a leap year.

The period from 1st January 2016 to 1st January 2017 includes the entire year 2016. Since 2016 was a leap year, the period contains 366 days.

Calculating the Number of Odd Days

Now we find the number of odd days in the 366 days between 1st January 2016 and 1st January 2017.

Number of odd days = \(366 \pmod{7}\)

\(366 \div 7 = 52\) with a remainder of \(2\).

So, there are 2 odd days in the period from 1st January 2016 to 1st January 2017.

Determining the Day of the Week for 1st January 2016

We know that 1st January 2017 was a Sunday. We are moving backward in time from 1st January 2017 to 1st January 2016. For every odd day, the day of the week shifts forward by one day when moving to a future date. Conversely, when moving to a past date, the day of the week shifts backward by one day for every odd day.

Since there are 2 odd days between 1st January 2016 and 1st January 2017, the day on 1st January 2016 was 2 days before Sunday.

• 1 day before Sunday is Saturday.
• 2 days before Sunday is Friday.

Therefore, if 1st January 2017 was Sunday, then 1st January 2016 was Friday.

Summary of Calculation

The calculation steps are:

1. Determine the period: From 1st Jan 2016 to 1st Jan 2017.
2. Identify if the period includes a leap year (2016 is a leap year).
3. Calculate the total number of days: 366 days.
4. Calculate the number of odd days: \(366 \pmod{7} = 2\).
5. Since we are going backward in time from 1st Jan 2017 (Sunday) to 1st Jan 2016, subtract the odd days from Sunday: Sunday - 2 days = Friday.

Final Answer Derivation

Based on our calculation, 1st January 2016 was a Friday.

To find the day going backward (from 2017 to 2016), we shift the day backward by the number of odd days:

Sunday \(\rightarrow\) Saturday (\(-1\) day) \(\rightarrow\) Friday (\(-2\) days)

Revision Table: Calendar Calculations

Additional Information: Calendar Odd Days

Understanding odd days is fundamental to solving calendar problems quickly. The number of odd days in a given period directly tells you the shift in the day of the week. For example:

• 0 odd days: Same day of the week.
• 1 odd day: Shift forward by 1 day.
• 2 odd days: Shift forward by 2 days.
• ... and so on.

When calculating the day for a date before a given date, you count backward the number of odd days. When calculating for a date after a given date, you count forward.

Remember that February in a leap year has 29 days instead of 28, adding the extra day. This extra day contributes to the additional odd day in a leap year.

Calendar problems often involve calculating odd days over centuries, years, months, and individual days. The principles remain the same: find the total number of days and determine the remainder when divided by 7.