Calendars Concepts

1. Finding the Day of the Week

What is the question? You are given a date and need to find which day of the week it falls on.

Formula:

$$\text{Day shift}=\text{Number of days}\mod 7$$

Since:

7 days = 1 complete week

only the remainder matters.

Example: If 1 January 1947 was Wednesday, what day was 15 August 1947?

Solution:

Days from 1 January to 15 August:

January     31
February    28
March       31
April       30
May         31
June        30
July        31
August      14
----------------
Total      226

We use 14 days in August because we are finding the shift from January 1 to August 15.

Now:

$$226\div7=32\text{ remainder }2$$

So the day moves forward by 2 days:

Wednesday → Thursday → Friday

Answer: Friday

---

2. Finding the Day After or Before a Given Number of Days

What is the question? A particular day is given, and you need to find the day after or before a certain number of days.

Formula:

$$\text{Day shift}=N\mod7$$

Example: If today is Monday, what day will it be after 100 days?

Solution:

$$100\mod7=2$$

So move 2 days after Monday:

Monday → Tuesday → Wednesday

Answer: Wednesday

---

3. Finding the Day Between Two Dates

What is the question? Find how many days or weeks lie between two given dates, or find the weekday of the second date when the first date is known.

Formula:

$$\text{Day shift} ================ \text{Total days}\mod7$$

Example: If 10 March is Monday, what day is 25 March in the same year?

Solution:

Days between the dates:

$$25-10=15$$

Now:

$$15\mod7=1$$

So the day moves forward by 1 day:

Monday → Tuesday

Answer: Tuesday

---

4. Leap Year

What is the question? Determine whether a particular year is a leap year. This is important because February has 29 days instead of 28.

Formula:

Divisible by 400       → Leap year
Divisible by 100       → Not a leap year
Divisible by 4         → Leap year
Otherwise              → Not a leap year

Example: Is 1900 a leap year? Is 2000 a leap year?

Solution:

1900:

1900 is divisible by 100
but not by 400

Therefore:

1900 → Not a leap year

2000:

2000 is divisible by 400

Therefore:

2000 → Leap year

Answer:

1900 → Not leap
2000 → Leap

---

5. Counting Odd Days in a Year

What is the question? Find how many days a year shifts the weekday of a date in the following year.

Formula:

For an ordinary year:

$$365\mod7=1$$

For a leap year:

$$366\mod7=2$$

Therefore:

Ordinary year → 1 odd day
Leap year     → 2 odd days

Example: If 1 January 2023 was Sunday, what day was 1 January 2024?

Solution:

2023 is an ordinary year.

So:

$$365\mod7=1$$

Move one day forward:

Sunday → Monday

Answer: Monday

---

6. Day on 1 January of a Future Year

What is the question? You are given the weekday of 1 January of one year and need to find the weekday of 1 January of a future year.

Formula:

$$\text{Total shift} ================== (\text{ordinary years})+(2\times\text{leap years})$$

Then:

$$\text{Shift}\mod7$$

Example: If 1 January 2020 was Wednesday, what day was 1 January 2025?

Solution:

Look at the years from 2020 to 2024:

2020 → Leap      → 2
2021 → Ordinary  → 1
2022 → Ordinary  → 1
2023 → Ordinary  → 1
2024 → Leap      → 2

Total:

$$2+1+1+1+2=7$$

Since:

$$7\mod7=0$$

there is no weekday shift.

Wednesday → Wednesday

Answer: Wednesday

---

7. Same Calendar / Repeating Calendar

What is the question? Find a future year in which the calendar is exactly the same as a given year.

For the calendars to be identical:

1. Both years must have the same type — both ordinary or both leap.
2. 1 January must fall on the same weekday.

Formula:

The accumulated odd days between the years must be:

$$0\pmod7$$

Example: Find a future year having the same calendar as 2023.

Solution:

2023 is an ordinary year.

Count the yearly shifts:

2023 → +1
2024 → +2  (leap)
2025 → +1
2026 → +1
2027 → +1
2028 → +2  (leap)
2029 → +1

Continue until the total shift becomes a multiple of 7 and the target year is also an ordinary year.

For aptitude questions, this is usually solved by accumulating odd days, rather than assuming a fixed 28-year cycle.

Remember:

Same weekday alone is not enough.
Year type must also match.

---

8. Number of Odd Days in a Century

What is the question? Find the weekday shift caused by 100, 200, 300, or 400 years.

Formula:

In 400 years:

303 ordinary years
97 leap years

Therefore:

$$303(1)+97(2)=497$$

And:

$$497\mod7=0$$

So:

400 years → 0 odd days

For common aptitude questions:

100 years → 5 odd days
200 years → 3 odd days
300 years → 1 odd day
400 years → 0 odd days

Example: How many odd days are there in 200 years?

Solution:

$$200\text{ years}\rightarrow3\text{ odd days}$$

Answer: 3 odd days

---

9. Finding the Number of Days Between Dates

What is the question? Find the exact number of days between two dates.

Formula:

Count:

Remaining days in starting month
+ complete months in between
+ days in ending month

Take care of:

February = 28 days normally
February = 29 days in a leap year

Example: How many days are there between 10 March and 25 March?

Solution:

$$25-10=15$$

Answer: 15 days

For weekday questions, then calculate:

$$15\mod7=1$$

So the weekday shifts by 1 day.

---

10. Finding the Day of the Week from a Known Reference Date

What is the question? A known date and its weekday are given. Use it to find the weekday of another date.

Formula:

$$\text{Target day} ================= \text{Reference day} + (\text{Days between dates}\mod7)$$

Example: 1 January 2025 was Wednesday. What day was 1 February 2025?

Solution:

January has 31 days:

$$31\mod7=3$$

Move 3 days from Wednesday:

Wednesday → Thursday → Friday → Saturday

Answer: Saturday

---

11. Day on a Particular Date in a Leap Year

What is the question? Find the weekday of a date in a leap year, especially when the date is after February 29.

Formula:

For a leap year:

February = 29 days

So when calculating a date after February, include the extra day.

Example: If 1 January 2024 was Monday, what day was 1 March 2024?

Solution:

January:

$$31\mod7=3$$

February in 2024:

$$29\mod7=1$$

Total shift:

$$3+1=4$$

Move 4 days from Monday:

Monday → Tuesday → Wednesday → Thursday → Friday

Answer: Friday

---

12. Counting the Occurrence of a Weekday

What is the question? Find how many Mondays, Tuesdays, etc. occur in a month or year.

Formula:

For a normal year:

$$365=52\times7+1$$

For a leap year:

$$366=52\times7+2$$

Therefore:

Ordinary year → 1 weekday occurs 53 times
Leap year     → 2 weekdays occur 53 times

Example: How many Sundays can an ordinary year have?

Solution:

365 = 52 weeks + 1 day

Every weekday occurs at least 52 times.

One weekday occurs an extra time.

So Sundays can occur:

52 or 53 times

Answer: 52 or 53

---

13. Day on the Last Day of a Month

What is the question? Find the weekday of the last day of a month when the weekday of the first day is known.

Formula:

$$\text{Last day} =============== \text{First day}+ (\text{Number of days}-1)\mod7$$

Example: If 1 April is Monday, what day is 30 April?

Solution:

April has 30 days.

Days to move:

$$30-1=29$$ $$29\mod7=1$$

So:

Monday → Tuesday

Answer: Tuesday