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Difficulty: MediumClocks and Calendars

A newly constructed high-speed railway network begins its first commercial operation on Sunday, January 15, 2096. The operating company schedules a comprehensive decade-review on the exact same date 10 years later, January 15, 2106. On what day of the week will this decade-review take place?

  1. A
    Thursday
  2. FridayAnswer
  3. C
    Saturday
  4. D
    Wednesday

Answer

Friday
The interval from January 15, 2096, to January 15, 2106, covers exactly 10 years. During this time, we cross February 29th in the years 2096 and 2104, making them leap years. Crucially, the year 2100 is a century year. Century years are only leap years if they are divisible by 400. Since 2100 is not divisible by 400, it is a normal year. This means we have 8 normal years and 2 leap years in this 10-year span. Calculating the odd days: (8 × 1) + (2 × 2) = 12 odd days. Dividing 12 by 7 gives a remainder of 5. Counting 5 days forward from Sunday lands exactly on Friday.

Step-by-Step Solution

1
Determine the total number of years in the given time frame.
The period from January 15, 2096, to January 15, 2106, is exactly 10 years.
This establishes the baseline period for calculating odd days.
2
Identify the leap years that fall within this 10-year period.
The leap years are 2096 (since February 29th is crossed) and 2104. The year 2100 is a century year not divisible by 400, so it is a normal year.
Century years must be divisible by 400 to be leap years, which is a critical rule for crossing the year 2100 boundary.
3
Calculate the total number of normal years and leap years.
Out of 10 years, there are 2 leap years and 8 normal years.
Leap years have 2 odd days and normal years have 1 odd day.
4
Calculate the total number of odd days for the 10-year period.
(8 normal years × 1 odd day) + (2 leap years × 2 odd days) = 8 + 4 = 12 odd days.
This translates the years into a continuous shift in the days of the week.
5
Find the net shift in the day of the week by finding the remainder when dividing by 7.
12 divided by 7 leaves a remainder of 5 odd days.
The days of the week repeat every 7 days.
6
Add the odd days to the starting day of the week.
Sunday + 5 days = Friday.
This determines the exact final day of the week for the scheduled event.

Key Concept

Calculation of odd days across century boundaries
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Topics

Clocks and Calendars

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