Calendar reasoning questions become much easier when you stop counting every day and start counting the remainder after complete weeks. Whether a question asks for a Monday in May, the weekday of a historical date, or the next year with an identical calendar, the basic tool is the same: a week has seven days.

This CrackTarget guide explains how to solve calendar-related questions in reasoning sections through worked examples, quick tables, common traps and a 25-question practice test from easy to very tough. Practise these ideas for aptitude and reasoning preparation wherever calendars appear in your examination syllabus.

Calendar convention: All dates and calculations below use the Gregorian calendar. For very early dates, this means applying the Gregorian rules mathematically; questions about a country’s actual historical calendar change require separate information. “After N days” means N days have elapsed. “Nth day” counts the starting date as day 1.

Jump to the part you need

• Solve the attached January–May 2026 question
• Odd days, leap years and the essential tables
• Worked examples of different problem types
• Shortcuts and points to remember
• Take the 25-MCQ calendar reasoning test
• Open the answer key and read detailed solutions
• Frequently asked questions

Solve the attached question: Monday in the second week of May 2026

Method 1: Move to the first day of May

Start from the known Thursday, 22 January. Count the days that elapse before 1 May. Do not count 22 January again: it is the starting point.

Part of the interval	Elapsed days
23–31 January	9
February 2026	28
March	31
April	30
1 May	1
Total	99 = 14 complete weeks + 1 day

Only the remainder matters. Thursday + 1 day = Friday, so 1 May 2026 is Friday. The first Monday is 4 May. Add one week: the next Monday is 11 May 2026. Therefore, the correct answer is option (d), 11th.

Method 2: Check the target date directly

From 22 January to 11 May, the elapsed days are 9 + 28 + 31 + 30 + 11 = 109. Since 109 = 7 × 15 + 4, move four weekdays forward from Thursday: Friday, Saturday, Sunday, Monday. This confirms that 11 May is Monday.

Mon	Tue	Wed	Thu	Fri	Sat	Sun
—	—	—	—	1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28	29	30	31

Wording trap: “Second week” can be used loosely in exam questions. If it means dates 8–14, the Monday is 11 May; if it means the second Monday of the month, it is also 11 May. A calendar grid can label its partial first row differently, so use the stated convention whenever a question provides one.

Calendar reasoning basics: the rules behind every shortcut

1. What are odd days?

Odd days are the days left after removing complete weeks. Divide the number of days by 7 and keep only the remainder. For example, 17 days = 14 + 3, so 17 days contain 3 odd days. Moving forward 17 days changes the weekday exactly as moving forward 3 days does.

Number of days	Complete weeks	Odd days
7	1	0
28	4	0
29	4	1
30	4	2
31	4	3
365	52	1
366	52	2

Quick example: If today is Monday, 100 days later will be Wednesday because 100 leaves remainder 2 on division by 7. If the question asks for the 100th day counting today as day 1, move only 99 days: the answer is Tuesday.

2. Ordinary years and leap years

• An ordinary or common year has 365 days, so it contributes 1 odd day.
• A leap year has 366 days, so it contributes 2 odd days.
• A non-century year is a leap year if divisible by 4.
• A century year must be divisible by 400 to be a leap year. Thus 2000 and 2400 are leap years; 1900 and 2100 are common years.

Use this order mentally: divisible by 400 → leap; otherwise divisible by 100 → common; otherwise divisible by 4 → leap; otherwise common. This order handles century exceptions without confusion.

3. Month lengths and their weekday shifts

Month length	Months	Shift to the next month’s 1st
31 days	Jan, Mar, May, Jul, Aug, Oct, Dec	+3
30 days	Apr, Jun, Sep, Nov	+2
28 days	February in a common year	0
29 days	February in a leap year	+1

Memory line: 30 days hath September, April, June and November. February has 28 or 29 days; the remaining months have 31. If 1 April is Wednesday, 1 May is Friday: April has 30 days, and 30 leaves remainder 2.

4. A consistent weekday-number system

Remainder	Weekday	Remainder	Weekday
0	Sunday	4	Thursday
1	Monday	5	Friday
2	Tuesday	6	Saturday
3	Wednesday	—	—

Use the same mapping throughout an absolute-date calculation. For relative questions, you can simply move forward or backward from the given day without assigning numbers.

Worked examples: the main types of calendar reasoning problems

Type 1: Find a weekday from a nearby known date

Example: If 10 January 2026 was Saturday, what day was 31 January? The gap is 31 − 10 = 21 days. Since 21 is exactly three weeks, the weekday remains Saturday. The shortcut is to subtract the date numbers when both dates belong to the same month and year.

Backward variation: If 20 July is Monday, 8 July is 12 days earlier. Reduce 12 to 5; move five days backward from Monday to Wednesday. Equivalently, moving backward 5 is moving forward 2.

Type 2: Cross one or more months

Example: If 1 March 2024 is Friday, find 1 April 2024. March has 31 days, so the shift is +3. Friday → Saturday → Sunday → Monday. February’s leap day does not enter this interval because it ends before 1 March.

For a longer gap, add the remaining days of the starting month, the complete intervening months, and the target date. Reduce each part modulo 7 if it keeps the arithmetic smaller. Never add the starting date itself to an elapsed-day count.

Type 3: Find the weekday of any date with a reliable formula

When no reference weekday is supplied, count complete years first. Let n = year − 1, M = days in the months before the target month, and d = target date number. Then calculate:

Why it works: each completed year contributes at least one odd day; each completed leap year adds one more. The subtraction removes century years that are not leap years, and the addition restores years divisible by 400. This formula uses the mathematical Gregorian anchor 1 January AD 1 = Monday.

Target month	M mod 7: common year	M mod 7: leap year
January	0	0
February	3	3
March	3	4
April	6	0
May	1	2
June	4	5
July	6	0
August	2	3
September	5	6
October	0	1
November	3	4
December	5	6

Worked example: 15 August 1947. Here n = 1946. Completed-year contribution = 1946 + 486 − 19 + 4 = 2417. Before August, the months contain 212 days, so add 212 + 15 = 227. Total = 2644. Since 2644 = 7 × 377 + 5, the remainder is 5: Friday.

Speed version: The completed-year total 2417 leaves remainder 2; the common-year August code is 2; the date 15 leaves remainder 1. Therefore 2 + 2 + 1 = 5, again Friday. Use either full day totals or month remainders consistently.

Type 4: The same date in the next year

Between the same month-and-date in consecutive years, the usual shift is +1 day. It becomes +2 only when 29 February lies inside the elapsed interval. Look at the actual interval instead of automatically looking at the label of the starting year.

Interval	Days elapsed	Weekday shift
26 Jan 2023 → 26 Jan 2024	365	+1
1 Mar 2023 → 1 Mar 2024	366	+2
1 Mar 2024 → 1 Mar 2025	365	+1

So if 26 January 2023 was Thursday, 26 January 2024 was Friday. But if 1 March 2023 was Wednesday, 1 March 2024 was Friday. The second interval contains 29 February 2024; the first does not.

Type 5: First, second, third or last occurrence of a weekday

Example: 1 May 2026 is Friday. The first Monday is date 4. Therefore the second, third and fourth Mondays are 11, 18 and 25. Once the first occurrence is known, repeatedly add 7.

Type 6: Count how many times a weekday occurs in a month

Every 28-day month contains exactly four of each weekday. A 29-day month adds one extra weekday; a 30-day month adds two consecutive extra weekdays; a 31-day month adds three. The extra sequence begins with the weekday of the 1st.

Example: A 31-day month begins on Tuesday. Its first 28 days contain four of every weekday; dates 29, 30 and 31 are Tuesday, Wednesday and Thursday. These three weekdays occur five times, and all others occur four times.

Type 7: Count 53 Sundays, Mondays or another weekday in a year

An ordinary year has 52 complete weeks plus one day. Only its starting weekday appears 53 times. A leap year has 52 complete weeks plus two days: its starting weekday and the next weekday each appear 53 times.

Example: A leap year begins on Saturday. Its extra weekdays are Saturday and Sunday, so it has 53 Saturdays and 53 Sundays. All other weekdays appear 52 times.

Probability variation: If a question explicitly treats all seven starting weekdays of a leap year as equally likely, 53 Sundays occur when 1 January is Saturday or Sunday. The probability is therefore 2/7. This is an exam model, not a claim that uniformly choosing real leap years from a Gregorian 400-year cycle gives equally likely starting weekdays.

Type 8: Find the next year with an identical calendar

Two complete calendars match only when both years start on the same weekday and both are common years or both are leap years. Matching 1 January alone is insufficient: February and the later months can differ.

Example: next calendar matching 2026. The shifts from 1 January 2026 to 1 January 2032 add the year lengths of 2026–2031. Their odd days total 1 + 1 + 2 + 1 + 1 + 1 = 7, so both years start on Thursday. But 2032 is leap and 2026 is common: reject 2032.

To reach 1 January 2037, add the year lengths of 2026–2036: 11 common-year base days plus extra days for leap years 2028, 2032 and 2036 = 14 odd-day units. This is divisible by 7, and 2037 is common. Therefore the next identical calendar is 2037. Checking earlier zero-remainder candidates confirms it is the next one.

Shortcut with a condition: A 28-year shift restores both weekday and year type when the four-year leap pattern remains uninterrupted across that interval. A skipped century leap day can break it. The Gregorian calendar’s full 400-year cycle is reliable because it has 97 leap years and 146,097 days, exactly divisible by 7.

Type 9: Leap-day birthdays

Example: Someone was born on Thursday, 29 February 2024. When will 29 February next fall on Thursday? Check only leap years. The weekdays of 29 February are Tuesday in 2028, Sunday in 2032, Friday in 2036, Wednesday in 2040, Monday in 2044, Saturday in 2048 and Thursday in 2052.

This question is about the next matching calendar date and weekday. It differs from asking how a person observes a birthday in a non-leap year. Around a century such as 2100, do not assume the same 28-year shortcut without checking.

Type 10: Reverse questions using weekday-date sums

Example: In a 31-day month, the sum of the dates of all Mondays is 80. Find the weekday of the 1st. If Monday occurred four times, the sum would be 4a + 42, where a is the first Monday, from 1 to 7; this gives at most 70. Therefore there must be five Mondays.

Their dates are a, a + 7, a + 14, a + 21 and a + 28. The sum is 5a + 70 = 80, so a = 2. The Mondays are 2, 9, 16, 23 and 30, which means the 1st is Sunday.

Type 11: Odd days in centuries

For a completed block beginning with AD 1, the first 100 years contain 24 leap years: 100 + 24 = 124 odd-day units, remainder 5. The first 200 and 300 years leave remainders 3 and 1. Four hundred years contain 97 leap years: 400 + 97 = 497, remainder 0.

Completed years from the anchor	Leap years	Odd days
100	24	5
200	48	3
300	72	1
400	97	0

Therefore 31 December of a Gregorian century year can be Friday, Wednesday, Monday or Sunday, according to the century’s position in the 400-year cycle. It cannot be Tuesday, Thursday or Saturday. Do not apply “100 years = 5 odd days” blindly to every arbitrary 100-year interval: an interval containing a divisible-by-400 leap century can have an extra leap day.

Type 12: Count leap years over a long interval

Define L(t) = floor(t/4) − floor(t/100) + floor(t/400). The number of leap years from year A through year B, including both endpoints, is L(B) − L(A − 1).

Example: 2001–2400 inclusive. L(2400) = 600 − 24 + 6 = 582; L(2000) = 500 − 20 + 5 = 485. Difference = 97 leap years. This method handles 2100, 2200 and 2300 correctly as common years while including leap year 2400.

Calendar reasoning shortcuts and points to remember

Question asks…	Fastest starting method
A nearby date’s weekday	Subtract dates; reduce the gap modulo 7
A date across months	Count elapsed days using actual month lengths
An absolute date without a given weekday	Completed-year formula + month remainder + date
Nth Monday, Sunday or another weekday	Find the first occurrence, then add multiples of 7
Five occurrences in a month	Use the 28 + 1/2/3 extra-days rule
Same calendar in another year	Match starting weekday AND year type
A range of leap years	Use L(B) − L(A − 1)
A reverse puzzle	Translate dates into a + 7k and use the constraints

• Reduce early: 31 + 30 + 31 has the same weekday effect as 3 + 2 + 3 = 8, or 1 odd day.
• Keep the direction clear: forward means add the remainder; backward means subtract it. Add 7 if the weekday number becomes negative.
• Separate elapsed and inclusive counting: from Monday to the following Monday is 7 elapsed days but 8 dates when both endpoints are counted.
• Look for 29 February inside the interval: the target year being leap is not enough on its own.
• Do not double-count the leap day: if your month table already uses a leap-year value, do not add another +1 after February.
• Reject an invalid date: 29 February 2100 and 31 April do not exist in the Gregorian calendar.
• A repeated weekday is not a repeated full calendar: check common versus leap status.
• Use options intelligently: dates exactly 7 apart have the same weekday. Once you locate one Monday, use that pattern.
• Check the convention: “second Monday” is precise; “second week” may need the question’s stated definition.
• Do not assume a probability model: use 1/7 or 2/7 only when the relevant seven weekday possibilities are treated as equally likely.

A useful exam routine: identify the type → write the interval → test for a leap day → reduce modulo 7 → map to the weekday → check the wording. If you make an error, record whether it came from a month length, a century exception, an extra starting day, or a wrong weekday mapping.

Calendar Reasoning Practice Test: 25 MCQs

Set A — Easy: Questions 1–5

Set B — Moderate: Questions 6–10

Set C — Advanced: Questions 11–15

Set D — Tough: Questions 16–20

Set E — Very Tough: Questions 21–25

Your response sheet

Q1–5	Q6–10	Q11–15	Q16–20	Q21–25
1 ___	6 ___	11 ___	16 ___	21 ___
2 ___	7 ___	12 ___	17 ___	22 ___
3 ___	8 ___	13 ___	18 ___	23 ___
4 ___	9 ___	14 ___	19 ___	24 ___
5 ___	10 ___	15 ___	20 ___	25 ___

Answer key: open after finishing the test

Tap to reveal the 25-question answer key

Question	Answer	Question	Answer
1	B	14	A
2	C	15	D
3	D	16	B
4	A	17	A
5	B	18	C
6	C	19	D
7	B	20	C
8	A	21	B
9	C	22	A
10	D	23	C
11	D	24	D
12	B	25	A
13	C	—	—

Detailed solutions to all 25 MCQs

Open each explanation after checking your responses. For every incorrect answer, identify the exact step that failed before attempting the question again.

Q1 — Show answer and explanation

Correct answer: B. Wednesday

100 = 98 + 2, so there are 2 odd days. Move two days forward from Monday: Wednesday. Because the wording says “after”, use 100 elapsed days, not 99.

Return to Question 1

Q2 — Show answer and explanation

Correct answer: C. 2400

2400 is divisible by 400. Century years 1900 and 2100 are not, while 2026 is not divisible by 4. Only 2400 is a leap year.

Return to Question 2

Q3 — Show answer and explanation

Correct answer: D. Saturday

The gap is 31 − 10 = 21 days = 3 complete weeks. The weekday remains Saturday.

Return to Question 3

Q4 — Show answer and explanation

Correct answer: A. 1

365 = 52 × 7 + 1. Therefore an ordinary year contains one odd day.

Return to Question 4

Q5 — Show answer and explanation

Correct answer: B. Wednesday

31 = 28 + 3. The first three weekdays, Tuesday, Wednesday and Thursday, occur five times. Of the options, only Wednesday qualifies.

Return to Question 5

Q6 — Show answer and explanation

Correct answer: C. 8 June

From 22 January to 1 June: 9 + 28 + 31 + 30 + 31 + 1 = 130 days. The remainder is 4, so Thursday moves to Monday. Thus 1 June is the first Monday and 8 June the second.

Return to Question 6

Q7 — Show answer and explanation

Correct answer: B. Friday

For completed years, 1946 + 486 − 19 + 4 = 2417. Add 212 days before August and date 15: 2644. Its remainder on division by 7 is 5, which means Friday in the Sunday = 0 mapping.

Return to Question 7

Q8 — Show answer and explanation

Correct answer: A. Monday

March has 31 days, producing a shift of 3. Friday + 3 = Monday. The leap day is outside this March-to-April interval.

Return to Question 8

Q9 — Show answer and explanation

Correct answer: C. Thursday

2026 is common. The gap between its 1 January and 31 December is 364 days, not 365. Since 364 is divisible by 7, both dates are Thursday.

Return to Question 9

Q10 — Show answer and explanation

Correct answer: D. 5

2028 is leap and 1 February is Tuesday. A 29-day month repeats its first weekday five times. Tuesdays fall on 1, 8, 15, 22 and 29.

Return to Question 10

Q11 — Show answer and explanation

Correct answer: D. Wednesday

Year 2023 contributes 1 odd day and leap year 2024 contributes 2. Total shift = 3: Sunday → Wednesday.

Return to Question 11

Q12 — Show answer and explanation

Correct answer: B. Friday

The interval is 365 days. It ends before 29 February 2024, so the shift is only +1. Thursday becomes Friday.

Return to Question 12

Q13 — Show answer and explanation

Correct answer: C. Friday

This interval includes 29 February 2024. Its length is 366 days, so move two weekdays from Wednesday to Friday.

Return to Question 13

Q14 — Show answer and explanation

Correct answer: A. Friday

In a 31-day month, exactly three consecutive weekdays occur five times, beginning with the weekday of the 1st. Friday, Saturday and Sunday imply the month began on Friday.

Return to Question 14

Q15 — Show answer and explanation

Correct answer: D. 2/7

A leap year has two extra consecutive weekdays. Sunday is included if 1 January is Saturday or Sunday: two favourable starting weekdays out of seven. The equal-likelihood assumption is explicitly part of the question.

Return to Question 15

Q16 — Show answer and explanation

Correct answer: B. 2037

2026 and 2032 start on Thursday, but 2032 is leap and cannot match common year 2026. From 1 January 2026 to 1 January 2037, 11 base days plus 3 leap days give 14, a multiple of 7. Year 2037 is common, making it the next full match.

Return to Question 16

Q17 — Show answer and explanation

Correct answer: A. 2052

Check leap years only. The February 29 weekdays are Tuesday (2028), Sunday (2032), Friday (2036), Wednesday (2040), Monday (2044), Saturday (2048), then Thursday (2052). No century exception intervenes.

Return to Question 17

Q18 — Show answer and explanation

Correct answer: C. 97

L(2400) − L(2000) = (600 − 24 + 6) − (500 − 20 + 5) = 582 − 485 = 97. Years 2100, 2200 and 2300 are common, while 2400 is leap.

Return to Question 18

Q19 — Show answer and explanation

Correct answer: D. Tuesday

The 100th day is 99 days after 1 January. Since 99 leaves remainder 1, Monday moves to Tuesday. The distinction between “100th day” and “after 100 days” is decisive.

Return to Question 19

Q20 — Show answer and explanation

Correct answer: C. Tuesday

The end-of-century weekdays repeat over the 400-year cycle: AD 100 ends Friday, 200 Wednesday, 300 Monday and 400 Sunday. Thus Tuesday is impossible; Sunday, Friday and Wednesday are possible.

Return to Question 20

Q21 — Show answer and explanation

Correct answer: B. Thursday

1899 contributes 365 days. January and February 1900 contribute 31 + 28 because 1900 is not a leap year. Total gap = 424 = 7 × 60 + 4. Sunday + 4 = Thursday.

Return to Question 21

Q22 — Show answer and explanation

Correct answer: A. Sunday

With four Mondays, the maximum possible sum is 4 × 7 + 42 = 70, so there must be five. If the first Monday is a, their sum is 5a + 70 = 80; hence a = 2. The 1st is therefore Sunday.

Return to Question 22

Q23 — Show answer and explanation

Correct answer: C. Tuesday

The two extra weekdays in a 30-day month must be Sunday and Monday, so the 1st is Sunday. The 17th is 16 days later: remainder 2. Sunday + 2 = Tuesday.

Return to Question 23

Q24 — Show answer and explanation

Correct answer: D. 2108

2096 is leap and begins on Sunday. The next leap candidates are 2104 and 2108 because 2100 is common. From 2096 to 2103 the shift is 8 + 1 = 9, so 2104 starts Tuesday. From 2096 to 2107 it is 12 + 2 = 14: 2108 starts Sunday and is leap. Therefore 2108 is the next full match.

Return to Question 24

Q25 — Show answer and explanation

Correct answer: A. 71

A 400-year cycle contains 400 × 365 + 97 = 146,097 days = 20,871 complete weeks, hence 20,871 Sundays. Assign 52 Sundays to each of the 400 years: 20,800. The remaining 71 Sundays occur one each in 71 years, because no year has more than 53 Sundays.

Return to Question 25

Use your score to choose the next revision task

Score out of 25	Suggested next step
0–10	Relearn odd days, month lengths and elapsed-day counting; redo Q1–10 slowly.
11–17	Practise leap-day intervals, nth weekdays and extra-day counting.
18–21	Review calendar repetition, century exceptions and reverse questions.
22–25	Repeat under a shorter time limit and explain the hardest solutions without notes.

These ranges are learning checkpoints for this practice set, not exam cut-offs. Aim first for correct reasoning, then reduce your time. A fast answer based on a misremembered shortcut will fail when the question changes one condition.

A short revision plan for mastering calendar questions

• Session 1: Learn common/leap rules and month shifts. Solve five nearby-date questions using actual elapsed days.
• Session 2: Work through the attached example and the absolute-date formula. Check one answer using a second method.
• Session 3: Practise weekday frequency, 53-weekday questions and calendar matching. Include a 2100 exception.
• Session 4: Attempt this 35-minute test. Review mistakes by cause and redo them without viewing the solutions.
• Session 5: Return after two days. Attempt the reverse and century questions first, then mix calendar questions with other reasoning topics.

Keep practising with CrackTarget

Use the same habit of understanding the method and reviewing errors across your preparation:

• Quantitative Aptitude Shortcuts: Time, Work and Speed with 30 MCQs
• Mock Test Analysis Guide: Review Errors for Maximum Improvement
• How to Use Previous-Year Questions Effectively
• Reasoning, Quant and GA Time-Management Strategy

Frequently asked questions about calendar reasoning

What is the easiest way to solve calendar reasoning questions?

For a relative-date question, count elapsed days and reduce the total modulo 7. For an absolute date without a supplied reference day, use the completed-year formula. Select the method that fits the information in the question.

Why does an ordinary year have one odd day?

An ordinary year has 365 days, equal to 52 complete weeks and one extra day. That extra day is its odd day.

Is 2100 a leap year?

No. Under Gregorian rules, a century year is leap only if divisible by 400. Year 2100 is not; year 2400 is.

Does the calendar always repeat after 28 years?

No. The shortcut depends on an uninterrupted four-year leap pattern across the interval. A skipped leap century can break it. Always check starting weekday and common/leap status; the full Gregorian cycle repeats after 400 years.

What is the answer to the attached May 2026 question?

11 May 2026, option (d). The given Thursday on 22 January leads to Friday on 1 May; Mondays then fall on 4, 11, 18 and 25 May.

How is the 100th day different from after 100 days?

When the starting date is day 1, the 100th day comes after 99 elapsed days. “After 100 days” means a full 100 days have passed.

Can an ordinary year have 53 Sundays?

Yes. It has 53 Sundays exactly when its 1 January is Sunday. A leap year has 53 Sundays when 1 January is Saturday or Sunday.

Calendar rule reference

The leap-year convention used here follows the U.S. Naval Observatory’s Gregorian leap-year rule. Worked examples, explanations and the practice test are prepared for CrackTarget. Date answers and calendar-repeat candidates were checked programmatically.