Calendar Questions and Answers

Answer: A

The correct statement is that 2nd February of that year is a Tuesday.

February can have a fifth Monday only in a leap year in which 1 February is a Monday. The Mondays then fall on the 1st, 8th, 15th, 22nd and 29th, and the fifth one is 29 February, the person's birthday. So 2 February is a Tuesday.

The other statements fail. With 1 February a Monday, the month starts on Monday, not Sunday, and its Sundays are the 7th, 14th, 21st and 28th, only four. The Mondays have dates 1, 8, 15, 22 and 29, mostly odd, so they do not all have even dates.

Answer: D

The 3rd of July falls on a Sunday.

Count the days from 2 June to 3 July. June has 30 days, so from 2 June to 2 July is 30 days, and one more day reaches 3 July: 31 days in all.

\[ 31 = 4 \times 7 + 3 \]

Four complete weeks bring the weekday back to Thursday, and the 3 extra days move it forward three places: Friday, Saturday, Sunday.

Thursday would need a gap that is a whole number of weeks (28 or 35 days). Friday and Saturday are what a count of 29 or 30 days gives, for instance by stopping at 2 July instead of 3 July.

Answer: D

September 11, 1919 is a Thursday.

Count the odd days (days beyond whole weeks) from May 5, 1911.

May 5, 1911 to May 5, 1919 is 8 years, including the leap years 1912 and 1916, so 8 + 2 = 10 odd days, which is 3 after removing a week. From May 5 to September 11, 1919: 26 days left in May, then 30, 31, 31 and 11:

\[ \begin{aligned} 26 + 30 + 31 + 31 + 11 &= 129 \\ 129 &= 18 \times 7 + 3 \end{aligned} \]

Total odd days: 3 + 3 = 6. Friday plus 6 days is Thursday. Forgetting the two leap days would give Tuesday, which is not offered.

Answer: A

The key gives Monday.

Days repeat every 7. Reading the question, as the key does, as the day that comes \(7^{9}\) days after today: \(7^{9}\) is an exact multiple of 7, so that many days is a whole number of weeks and the weekday is unchanged.

\[ 7^{9} \equiv 0 \pmod 7 \]

The point of the question is that any power of 7 leaves remainder 0 on division by 7, so no counting is needed. Wednesday, Saturday and Tuesday would each need a remainder of 2, 5 or 1 days.

Answer: C

The father's birthday falls on Thursday.

Both birthdays (September and December) come after 29 February, so they shift by exactly the same number of weekdays from year to year. An ordinary year moves a date on by 1 day and a leap year by 2.

\[ \begin{aligned} \text{2017 to 2020} &= 1 + 1 + 2 \\ &= 4\ \text{days} \end{aligned} \]

Sunday moved on 4 days is Thursday, which agrees with Atul's birthday. The father's birthday, also a Sunday in 2017, moves by the same 4 days, so it too falls on a Thursday.

Wednesday comes from counting 2020 as an ordinary year (a shift of 3), and Friday from adding an extra day. The months matter only to show both dates fall after February.

Answer: A

The next Tuesday anniversary falls in 2020.

A date moves forward one weekday each year, and two when a 29 February falls in between. Starting from Tuesday, 15 September 2015:

• 2016: +2 (leap year), Thursday
• 2017: +1, Friday
• 2018: +1, Saturday
• 2019: +1, Sunday
• 2020: +2 (leap year), Tuesday

The total shift is 7 days, a whole week, so 15 September 2020 is again a Tuesday. 2021 is a year too many, and 2019 falls on a Sunday.

Answer: A

The Sunday is 20 March 2050.

From 15 March 2025 to 15 March 2050 is 25 years. Each ordinary year moves the weekday on by 1, and each 29 February in between adds 1 more. The leap days fall in 2028, 2032, 2036, 2040, 2044 and 2048, which is 6.

\[ \begin{aligned} \text{shift} &= 25 + 6 = 31 \\ 31 &\equiv 3 \pmod 7 \end{aligned} \]

So 15 March 2050 falls 3 days after Saturday, on a Tuesday. The following Sunday is 15 + 5 = 20 March.

Forgetting the leap days gives a shift of 25, which leaves 4, making 15 March a Wednesday and pointing wrongly to the 19th.

Answer: C

January 1st next falls on a Sunday in 2034.

Each ordinary year moves the weekday of January 1 forward by one day, and a leap year by two. Starting from Sunday in 2023:

• 2024 Monday; 2025 Wednesday (2024 is a leap year); 2026 Thursday; 2027 Friday; 2028 Saturday
• 2029 Monday (2028 is a leap year); 2030 Tuesday; 2031 Wednesday; 2032 Thursday
• 2033 Saturday (2032 is a leap year); 2034 Sunday

The list shows that 2029 starts on a Monday and 2030 on a Tuesday; continuing, 2035 is Monday, 2036 Tuesday and 2037 Thursday (2036 is a leap year), so none of the other options works. Ignoring leap years would wrongly give 2030, seven years on.

Answer: D

The fourth Tuesday falls on the 26th.

The 30th is a Saturday, so counting back, the 26th is a Tuesday (Saturday 30, Friday 29, Thursday 28, Wednesday 27, Tuesday 26). The Tuesdays of the month are then 26, 19, 12 and 5, that is the 5th, 12th, 19th and 26th.

Since the first Tuesday is the 5th, the fourth is the 26th. 20 and 21 are not Tuesdays in this month, and 25 is a Monday.

Answer: C

There are 24 working days.

If the 29-day month begins on a Saturday, the Saturdays fall on the 1st, 8th, 15th, 22nd and 29th, and the Sundays on the 2nd, 9th, 16th and 23rd.

The holidays are the second Saturday (the 8th) and the four Sundays, 5 days in all, so

\[ 29 - 5 = 24 \]

25 is what you get by counting only the four Sundays and forgetting the second Saturday. 23 and 22 count six or seven holidays, as if more than one Saturday were off; only the second Saturday is a holiday here.