NCERT Solutions Ganita Prakash Chapter 3 In-text Questions — Clock and Calendar Numbers

Book page 64 Updated on2026-09-05

Q1.

On the usual 12-hour clock, there are timings with different patterns. For example, 4:44, 10:10, 12:21. Try and find out all possible times on a 12-hour clock of each of these types.

Answer

There are three different patterns here. Take them one at a time.

Type 1 — the same digit all through (like 4:44)

1:11 2:22 3:33 4:44 5:55 11:11

6:66 and beyond are impossible, because minutes can only go up to 59.

Type 2 — the minutes repeat the hour (like 10:10)

01:01 02:02 03:03 04:04 05:05 06:06
07:07 08:08 09:09 10:10 11:11 12:12

That is 12 such times.

Type 3 — palindromic times, reading the same both ways (like 12:21)

01:10 02:20 03:30 04:40 05:50
10:01 11:11 12:21

06:60 is impossible, so the list stops at 05:50.

Did you notice? 11:11 belongs to all three lists — it is the only time that is repeated, palindromic and made of a single digit.

Q2.

Manish has his birthday on 20/12/2012 where the digits ‘2’, ‘0’, ‘1’, and ‘2’ repeat in that order. Find some other dates of this form from the past.

Answer

In such a date the year simply repeats the day and the month: 20/12/2012. So pick any day and month, join them, and that is the year.

Date	Day & month	Year formed
20/01/2001	20 and 01	2001
20/04/2004	20 and 04	2004
20/06/2006	20 and 06	2006
20/11/2011	20 and 11	2011
19/11/1911	19 and 11	1911
18/12/1812	18 and 12	1812
17/09/1709	17 and 09	1709

The rule: for a date in the 2000s the day must be 20 (so that the year starts with 20), and the month can be anything from 01 to 12. For older centuries the day gives the century — 19 gives 19xx, 18 gives 18xx, and so on.

Q3.

His sister, Meghana, has her birthday on 11/02/2011 where the digits read the same from left to right and from right to left. Find all possible dates of this form from the past.

Answer

Write the date as eight digits: 1 1 0 2 2 0 1 1. For a palindrome the year must be the day and month written backwards.

In our own century the year has to start with 2 and then 0, so the month must be 02 (February). That gives all the palindromic dates of the 2000s:

Date	Written out	Read backwards
10/02/2001	10022001	10022001 ✓
20/02/2002	20022002	20022002 ✓
01/02/2010	01022010	01022010 ✓
11/02/2011	11022011	11022011 ✓
21/02/2012	21022012	21022012 ✓
02/02/2020	02022020	02022020 ✓
12/02/2021	12022021	12022021 ✓
22/02/2022	22022022	22022022 ✓

These eight are all the palindromic dates so far in the 21st century — and every one of them falls in February!

Did you know? Older examples exist too: 11/11/1111 and 12/11/1121. The next palindromic date after 22/02/2022 is 03/02/2030.

Q4.

Jeevan was looking at this year’s calendar. He started wondering, “Why should we change the calendar every year? Can we not reuse a calendar?” But, will any year’s calendar repeat again after some years? Will all dates and days in a year match exactly with that of another year?

Answer

Yes — calendars do repeat, and an old calendar can be reused.

An ordinary year has 365 days = 52 weeks + 1 extra day, so the same date shifts forward by one weekday next year. A leap year has 366 days = 52 weeks + 2 extra days, so it shifts by two.

1 January 2014 was a Wednesday → 1 January 2025 was also a Wednesday
Both years are ordinary years, so the whole calendar of 2014 matches that of 2025.

An ordinary year’s calendar usually returns after 6, 11 or 28 years — for example 2018 and 2029 have exactly the same calendar.
A leap year’s calendar returns after 28 years — 2024’s calendar will be reusable in 2052.

Two things must match: (1) the two years must begin on the same weekday, and (2) both must be leap years, or both ordinary years. Only then will every date fall on the same day.

Try This: find an old calendar at home and check which future year it can be used for.

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