Introduction / Context:
This conceptual calendar problem asks about the relationship between dates in March and November of the same year. Specifically, it claims that any date in March and the same date number in November fall on the same day of the week. To answer correctly, we must understand how many days lie between 1 March and 1 November and how this number behaves modulo 7.
Given Data / Assumptions:
- We consider a single year under the Gregorian calendar.
- We compare dates such as 1 March and 1 November, 5 March and 5 November, and so on.
- We assume normal month lengths: March 31, April 30, May 31, June 30, July 31, August 31, September 30, October 31, November 30.
- The year may be leap or non leap; we must see whether that matters after February is over.
Concept / Approach:
If the difference in days between 1 March and 1 November is a multiple of 7, then 1 March and 1 November fall on the same weekday. Once that is true, adding the same number of days to both dates (for example adding 4 days to go from 1 March to 5 March and from 1 November to 5 November) keeps the weekday alignment. Therefore, the date numbers throughout March and November will share the same weekday. We only need to show that the number of days between 1 March and 1 November is divisible by 7.
Step-by-Step Solution:
Consider the months from March to October inclusive, as they lie between March and November.
Days in each month: March 31, April 30, May 31, June 30, July 31, August 31, September 30, October 31.
From 1 March to 1 April is 31 days, 1 April to 1 May is 30 days, and so on.
Total days from 1 March to 1 November = 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31.
Compute the sum: 31 + 30 = 61, plus 31 = 92, plus 30 = 122, plus 31 = 153, plus 31 = 184, plus 30 = 214, plus 31 = 245.
245 divided by 7 gives 35 exactly, with no remainder.
Therefore, 1 March and 1 November always fall on the same weekday.
Adding k days to both dates keeps the weekday same, so k March and k November also fall on the same weekday for any valid date number k.
Verification / Alternative check:
Check with an actual year, for example 2020 or 2021, using a calendar to see that 3 March and 3 November, 10 March and 10 November, and so on, match in weekday.
Because the counting starts from March, the leap day in February does not affect this relationship.
Thus the property holds in both leap and non leap years.
Why Other Options Are Wrong:
Option B (Not same day): Contradicted by the fact that the difference is exactly 245 days, a multiple of 7.
Option C (Next day): Would require the difference to be 1 modulo 7, not 0.
Option D (Previous day): Would correspond to a difference of 6 modulo 7, again not 0.
Option A (Same day): Correct, as proven by the 245 day difference.
Common Pitfalls:
Some learners miscount the days in one or more months, breaking the multiple of 7 property.
Another mistake is to include February in the calculation, even though we start from March.
Ignoring the modulo 7 idea and trying to track weekdays month by month without structure can also lead to confusion.
Final Answer:
Any date in March falls on the same weekday as the corresponding date in November of that year.
Discussion & Comments