Clocks and Calendar Problems — MCQs with Answers (Set 1)

A calendar repeats only when the years span the same number of odd days and neither has a leap-year mismatch in between. 2025 is not a leap year and starts on a Wednesday. Counting odd days through 2026-2030 (three normal years and one leap year, 2028), 2031 also starts on a Wednesday and is not a leap year, so it matches 2025 exactly.

A correct clock’s hands coincide every 720/11 = 65 5/11 minutes of its own face-time. Here this happens every 65 real minutes, so the clock runs fast. When the clock shows 12 hours (720 minutes) have passed, real elapsed time is only 715 minutes, or 11 hours 55 minutes. So the actual time is 7:55 PM.

A clock’s mirror image follows one rule: mirror time = 11:60 minus the actual time. Here the given time is 3:50. 11:60 minus 3:50 works out to 8:10. So the answer is 8:10. This rule works both ways: applying it twice returns the original time.

A clock’s mirror image follows one rule: mirror time = 11:60 minus the actual time. Here the given time is 1:05. 11:60 minus 1:05 works out to 10:55. So the answer is 10:55. This rule works both ways: applying it twice returns the original time.

Take 23 August 2020 as the known reference day, a Sunday. The span to 23 August 2029 covers 9 one-year gaps. Each gap runs from 23 August of one year to 23 August of the next. Check which gaps pass through a 29 February. The gap starting in 2023 passes through February 2024, a leap year. The gap starting in 2027 passes through February 2028, also a leap year. No other gap in this span passes through a leap February. So 2 of the 9 gaps carry an extra day, and 7 gaps are ordinary. Each ordinary gap contributes 1 odd day, and each leap-affected gap contributes 2. The total is 7 times 1, plus 2 times 2, which equals 7 plus 4, or 11 odd days. Divide 11 by 7. That gives 1 full week, with a remainder of 4 days. Move 4 days forward from Sunday. Sunday plus 4 lands on Thursday, after passing Monday, Tuesday, and Wednesday. So 23 August 2029 falls on a Thursday.

By the standard convention used in these problems, 1 January of year 1 CE is a Monday. From year 1 to year 2101 is a span of 2100 years. Break 2100 into complete 400-year blocks first. That gives 5 complete blocks of 400 years, covering 2000 years. Each complete 400-year block contributes exactly 0 odd days. That leaves 100 years still to account for. A 100-year block, counted from the start of a century, contributes exactly 5 odd days. So the total odd days across all 2100 years is 0 plus 5, which is 5. Move 5 days forward from Monday. Monday plus 5 lands on Saturday, after passing Tuesday, Wednesday, Thursday, and Friday. So 1 January 2101 falls on a Saturday.

By the standard convention used in these problems, 1 January of year 1 CE is a Monday. From year 1 to year 1901 is a span of 1900 years. Break 1900 into complete 400-year blocks first. That gives 4 complete blocks of 400 years, covering 1600 years. Each complete 400-year block contributes exactly 0 odd days, since it holds exactly 97 leap years. That leaves 300 years still to account for. A 300-year block, counted from the start of a century, contributes exactly 1 odd day. So the total odd days across all 1900 years is 0 plus 1, which is 1. Move 1 day forward from Monday. Monday plus 1 is Tuesday. So 1 January 1901 falls on a Tuesday.

Take 15 March 2023 as the known reference day, a Wednesday. Check each following year’s gap for odd days, one year at a time. The gap into 2024 passes through February 2024, a leap year, adding 2 odd days. Running total after 2024: 2 odd days, so 15 March 2024 is a Friday. The gap into 2025 is ordinary, adding 1 odd day, for a running total of 3. The gap into 2026 is also ordinary, adding 1 more, for a running total of 4. The gap into 2027 is ordinary too, adding 1 more, for a running total of 5. The gap into 2028 passes through February 2028, a leap year, adding 2 odd days. That brings the running total to 5 plus 2, or 7 odd days. Divide 7 by 7. That gives exactly 1 full week, with a remainder of 0. A remainder of 0 means the day matches the reference day again. So 15 March next falls on a Wednesday in the year 2028.

Take 9 November 2021 as the known reference day, a Tuesday. The span to 9 November 2026 covers 5 one-year gaps. Each gap runs from 9 November of one year to 9 November of the next. Check which gaps pass through a 29 February. Only the gap starting in 2023 passes through February 2024, a leap year. The other 4 gaps do not pass through a leap February. So 1 of the 5 gaps carries an extra day, and 4 gaps are ordinary. Each ordinary gap contributes 1 odd day, and the leap-affected gap contributes 2. The total is 4 times 1, plus 1 times 2, which equals 4 plus 2, or 6 odd days. Divide 6 by 7. The remainder is still 6 days, since 6 is less than 7. Move 6 days forward from Tuesday. That lands on Monday, one day short of a full week. So 9 November 2026 falls on a Monday.

Take 14 July 2019 as the known reference day, a Sunday. The span to 14 July 2026 covers 7 one-year gaps. Each gap runs from 14 July of one year to 14 July of the next. Check which gaps pass through a 29 February. The gap starting in 2019 passes through February 2020, a leap year. The gap starting in 2023 passes through February 2024, also a leap year. No other gap in this span passes through a leap February. So 2 of the 7 gaps carry an extra day, and 5 gaps are ordinary. Each ordinary gap contributes 1 odd day, and each leap-affected gap contributes 2. The total is 5 times 1, plus 2 times 2, which equals 5 plus 4, or 9 odd days. Divide 9 by 7. That gives 1 full week, with a remainder of 2 days. Move 2 days forward from Sunday. Sunday plus 2 is Tuesday. So 14 July 2026 falls on a Tuesday.

Take 20 May 2025 as the known reference day, a Tuesday. The span from 20 May 2025 to 20 May 2026 passes through February 2026. The year 2026 does not divide evenly by 4, so it is not a leap year. February 2026 has only 28 days, so no extra day is added. The full span is therefore a normal 365 days. Divide 365 by 7. That gives 52 full weeks, with a remainder of 1 day. Move 1 day forward from Tuesday. Tuesday plus 1 is Wednesday. So 20 May 2026 falls on a Wednesday.

Take 1 January 2028 as the known reference day, a Saturday. The year 2028 divides evenly by 4 and is not a century year, so it is a leap year. A leap year has 366 days in total. The gap from 1 January to 31 December is one day short of that total. That gap is 366 minus 1, which equals 365 days. Divide 365 by 7. That gives 52 full weeks, with a remainder of 1 day. Move 1 day forward from Saturday. Saturday plus 1 is Sunday. So 31 December 2028 falls on a Sunday.

Take 1 January 2026 as the known reference day, a Thursday. The year 2026 is not a leap year, so it has 365 days in total. The gap from 1 January to 31 December is one day short of the full year. That gap is 365 minus 1, which equals 364 days. Divide 364 by 7. That gives exactly 52 full weeks, with a remainder of 0 days. A remainder of 0 means the day of the week does not change. So 31 December 2026 also falls on a Thursday.

Count the days between 10 February 2028 and 22 July 2028. The year 2028 divides evenly by 4 and is not a century year, so it is a leap year. February 2028 therefore has 29 days, so 19 days remain after the 10th. March contributes 31 days, April contributes 30, and May contributes 31. June contributes 30 days, and July contributes 22 days, up to the target date. Adding these gives 19 plus 31 plus 30 plus 31 plus 30 plus 22, which is 163 days. Divide 163 by 7. That gives 23 full weeks, with a remainder of 2 days. This remainder of 2 is the number of odd days between the two dates.

Count the days between 12 March 2025 and 5 August 2025. March contributes 19 remaining days, since 31 minus 12 equals 19. April contributes 30 days, May contributes 31, and June contributes 30. July contributes 31 days, and August contributes 5 days, up to the target date. Adding these gives 19 plus 30 plus 31 plus 30 plus 31 plus 5, which is 146 days. The year 2025 is not a leap year, and February is not in this span anyway. Divide 146 by 7. That gives 20 full weeks, with a remainder of 6 days. This remainder of 6 is the number of odd days between the two dates.

Take 20 January 2024 as the known reference day, a Saturday. Count the days from 20 January to 4 April 2024. January contributes 11 remaining days, since 31 minus 20 equals 11. The year 2024 divides evenly by 4 and is not a century year, so it is a leap year. February 2024 therefore contributes 29 days, not 28. March contributes 31 days, and April contributes 4 days, up to the target date. Adding these gives 11 plus 29 plus 31 plus 4, which is 75 days. Divide 75 by 7. That gives 10 full weeks, with a remainder of 5 days. Move 5 days forward from Saturday. Saturday plus 5 lands on Thursday, after passing Sunday, Monday, Tuesday, and Wednesday. So 4 April 2024 falls on a Thursday.

Take 14 January 2026 as the known reference day, a Wednesday. Count the days from 14 January to 2 April 2026. January contributes 17 remaining days, since 31 minus 14 equals 17. February 2026 is not a leap year, so it contributes 28 days. March contributes 31 days, and April contributes 2 days, up to the target date. Adding these gives 17 plus 28 plus 31 plus 2, which is 78 days. Divide 78 by 7. That gives 11 full weeks, with a remainder of 1 day. Move 1 day forward from Wednesday. Wednesday plus 1 is Thursday. So 2 April 2026 falls on a Thursday.

Take 9 January 2027 as the known reference day, a Saturday. Count the days from 9 January to 14 June 2027. January contributes 22 remaining days, since 31 minus 9 equals 22. February 2027 is not a leap year, so it contributes 28 days. March contributes 31 days, April contributes 30, and May contributes 31. June contributes 14 days, up to the target date. Adding these gives 22 plus 28 plus 31 plus 30 plus 31 plus 14, which is 156 days. Divide 156 by 7. That gives 22 full weeks, with a remainder of 2 days. Move 2 days forward from Saturday. Saturday plus 1 is Sunday, plus 2 is Monday. So 14 June 2027 falls on a Monday.

Start from the known fact: today is a Saturday. The target is 24 days from today. Divide 24 by 7. That gives 3 full weeks, with a remainder of 3 days. This remainder of 3 is the odd-day shift. Move 3 days forward from Saturday. Saturday plus 1 is Sunday, plus 2 is Monday, plus 3 is Tuesday. So 24 days from now will be a Tuesday.

Start from the known fact: yesterday was a Sunday. If yesterday was Sunday, today must be Monday. Tomorrow, one day after today, must be Tuesday. The day after tomorrow is one more day after that. One day after Tuesday is Wednesday. So the day after tomorrow will be a Wednesday.

Take 25 December 2026 as the known reference day, a Friday. The target date, 14 December 2026, is 11 days earlier. Divide 11 by 7. That gives 1 full week, with a remainder of 4 days. This remainder of 4 is the odd-day shift, counted backward. Both dates fall in December, so no leap-year rule applies here. Move 4 days backward from Friday. Friday minus 1 is Thursday, minus 2 is Wednesday, minus 3 is Tuesday, minus 4 is Monday. So 14 December 2026 fell on a Monday.

Take 4 March 2026 as the known reference day, a Wednesday. The target date, 21 March 2026, is 17 days later. Divide 17 by 7. That gives 2 full weeks, with a remainder of 3 days. This remainder of 3 is the odd-day shift. Both dates fall in March, so no leap-year rule applies here. Move 3 days forward from Wednesday. Wednesday plus 1 is Thursday, plus 2 is Friday, plus 3 is Saturday. So 21 March 2026 falls on a Saturday.

A year is a leap year if it divides evenly by 4. A century year only stays a leap year if it also divides by 400. Check 2000: it divides by 4 and by 400. So 2000 is a leap year. Check 2400: it also divides by 4 and by 400. So 2400 is a leap year too. Check 2016: it divides by 4, and it is not a century year. So 2016 is a leap year as well. Check 1800: it divides by 4, but 1800 divided by 400 leaves a remainder. So 1800 fails the century test. The year that is not a leap year is 1800.

A year is a leap year if it divides evenly by 4. A century year is the exception to that rule. A century year must divide evenly by 400 to stay a leap year. Check 1900: it divides by 4, but 1900 divided by 400 leaves a remainder. So 1900 is not a leap year. Check 2100: it also divides by 4, but not by 400. So 2100 is not a leap year either. Check 2200: same story, divisible by 4 but not by 400. So 2200 is also not a leap year. Check 2032: it divides by 4, and it is not a century year at all. So 2032 is a leap year. The leap year among the four options is 2032.

A water image time follows this rule: water time equals 6:00 minus the actual time. At 4:10, this subtraction needs borrowing, since 10 is more than 0. Rewrite 6:00 as 5 hours and 60 minutes. Subtract the hours: 5 minus 4 equals 1. Subtract the minutes: 60 minus 10 equals 50. So the water image reads 1:50.