1 st January 2018 was a Monday. Which of the following years will also start on a Monday?
2024
Understanding How Calendar Days Repeat
This question asks us to find when the calendar year will start on a Monday again, given that January 1st, 2018 was a Monday. This is a common type of calendar problem that involves understanding 'odd days'.
What are Odd Days?
An odd day is the number of days remaining after dividing the total number of days by 7 (the number of days in a week). When we move from one date to another, the day of the week shifts forward by the number of odd days between those two dates.
- A normal year has 365 days. \\(365 \div 7 = 52\\) weeks and \\(1\\) day remaining. So, a normal year has \\(1\\) odd day.
- A leap year has 366 days. \\(366 \div 7 = 52\\) weeks and \\(2\\) days remaining. So, a leap year has \\(2\\) odd days.
Leap years occur every 4 years, except for century years not divisible by 400. The years we are considering (2018, 2019, 2020, 2021, 2022, 2023, 2024) include the leap year 2020 and 2024.
Finding the Repeating Year
The starting day of a year will repeat when the total number of odd days accumulated from the reference year (2018) to the end of the year preceding the target year is a multiple of 7. We need to calculate the cumulative odd days year by year starting from 2018.
| Year | Type of Year | Odd Days in the Year | Cumulative Odd Days (from end of 2018) | Day on Jan 1st of next year (Relative to Mon + Odd Days) |
|---|---|---|---|---|
| 2018 | Normal | 1 | 1 (after end of 2018) | Monday + 1 = Tuesday (Jan 1, 2019) |
| 2019 | Normal | 1 | 1 (from 2018) + 1 (from 2019) = 2 (after end of 2019) | Monday + 2 = Wednesday (Jan 1, 2020) |
| 2020 | Leap | 2 | 2 + 2 (from 2020) = 4 (after end of 2020) | Monday + 4 = Friday (Jan 1, 2021) |
| 2021 | Normal | 1 | 4 + 1 (from 2021) = 5 (after end of 2021) | Monday + 5 = Saturday (Jan 1, 2022) |
| 2022 | Normal | 1 | 5 + 1 (from 2022) = 6 (after end of 2022) | Monday + 6 = Sunday (Jan 1, 2023) |
| 2023 | Normal | 1 | 6 + 1 (from 2023) = 7 (after end of 2023) | Monday + 7 = Monday (Jan 1, 2024) |
As shown in the table, the cumulative number of odd days reaches 7 after the end of the year 2023. Since 7 is a multiple of 7, the day of the week on January 1st, 2024 will be the same as January 1st, 2018, which was a Monday.
Therefore, the year 2024 will also start on a Monday.
Revision Table: Calendar Odd Days Calculation
Here is a quick summary of odd days:
| Period | Number of Days | Odd Days (Days % 7) |
|---|---|---|
| Normal Year | 365 | 1 |
| Leap Year | 366 | 2 |
| 100 years (Normal Century) | 76 Normal + 24 Leap years | 76*1 + 24*2 = 76 + 48 = 124. \\(124 \div 7 = 17\\) remainder \\(5\\). 5 odd days. |
| 400 years | Sum of odd days for 4 normal centuries + 1 extra day for leap century = \\(4 \times 5 + 1 = 21\\). \\(21 \div 7 = 3\\) remainder \\(0\\). 0 odd days. | 0 |
Additional Information on Calendar Concepts
Understanding the calendar system involves a few key concepts:
- Leap Years: A year is a leap year if it is divisible by 4, unless it is a century year. Century years are leap years only if they are divisible by 400. For example, 2000 was a leap year (divisible by 400), but 1900 was not (divisible by 100 but not 400). This rule helps synchronize the calendar with the Earth's orbit around the sun.
- Cycle of Days: The day of the week repeats every 7 days. Therefore, the day of the week for a particular date repeats after a cycle of years where the total number of odd days is a multiple of 7. The cycle of the calendar repeating itself can be 6, 11, 28, or 400 years, depending on the position relative to leap years.
- Calculating Day of the Week: The concept of odd days is fundamental in calculating the day of the week for a given date. By finding the total number of odd days from a known reference date, we can determine the day of the week for any other date.
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