Calendar Problems: Concepts & Tricks

Introduction
Most students face difficulty in solving Calendar math problems. So here we try to provide a simplified procedure to solve calendar reasoning problems.
Let us begin with the basics. We know that in an ordinary year there are 365 days, which means 52 × 7 + 1, or 52 weeks and one day. This additional day, we call an odd day. If 1st January of this year is on Sunday, then 1st January next year will be exactly 52 full weeks and a day after that – so on a Monday.
This is all right as long as the year is not a leap year. The Earth actually completes 1 orbit around the Sun in more than 365 days, i.e in 365 Days 5 Hours 48 minutes and 45 seconds or takes approximately 6 hours more. A leap year occurs every 4 years to adjust for the 1/4th day, 6 x 4= 24 hours, so every 4th year has 366 days (or 2 odd days). And as far as the few odd minutes of the orbit time are concerned, well every 100 years starting 1 AD, the year is declared to be a non–leap year, but every 4th century is a leap year. So any year divisible by 400 will be a leap year e.g. : 1200, 1600 and 2000. The years 1800, 1900 will be non leap years.
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Concept of ‘odd-days’
The concept of odd days is very important in determining the days of the week. Let us look at how many odd days will there be in a century – i.e. 100 years. There will be 24 leap years and 76 non–leap years. As studied earlier each leap year and 2 odd days and each non-leap year has 1 odd day. Therefore, there will be 24 × 2 + 76 × 1 = 124 total odd days. Since 7 odd days make a week, to find out the net odd days, divide 124 by 7. The remainder is 5 – this is the number of odd days in a century.
Calendar Aptitude Tricks
You can remember the following points relating to the concepts of calendar:
  • 100 years give us 5 odd days as calculated above.
  • 200 years give us 5 x 2 = 10 – 7 (one week) 3 odd days.
  • 300 years give us 5 x 3 = 15 – 14 (two weeks) 1 odd day.
  • 400 years give us {5 x 4 + 1 (leap century)} – 21} (three weeks) 0 odd days.
  • Month of January gives us 31 – 28 = 3 odd days.
  • Month of February gives us 28 – 28 = 0 odd day in a normal year and 1 odd day in a leap year and so on for all the other months.
In total first six months i.e. January to June give us 6 odd days in a normal year and 7 – 7 = 0 odd days in a leap year. This is going to help, when you want to find a day, which is after 30th June. In total first nine months i.e. January to September give us 0 odd days in a normal year and 1 odd day in a leap year. Sometimes a reference date might be given to you. This makes your task easier, as you then start counting odd days only from that day.
When you count from the beginning i.e. 1st January, 0001
  • 1 odd day mean – Monday
  • 2 odd days mean – Tuesday
  • 3 odd days mean – Wednesday and so on 6 odd days means Saturday.
Now let us solve an illustration:
Illustration: Any date in March is the same day of the week as corresponding date in of the same year.
A. October
B. November
C. June
D. September
Answer: Option 2
Solution: 2 months have same corresponding days if number of odd days between these 2 months is 0. i.e. the total number of days are divisible by 7.
Now, between March and November, total number of days = 245.
Hence number of odd days =(245/7) 0.
So these 2 months have exactly same calendar.
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