Try and find out all possible times on a 12-hour clock of each of these types.

Question

Homi · Accepted Answer

We will look for special patterns in the digits of the time.

Step 1 — Single digit repeating times

First, let us find times where the hour is a single digit. The minutes will be that same digit repeated twice. A 12-hour clock shows hours from 1 to 12. Minutes go from 00 to 59.

Let the hour be a single digit, for example, 1. The minutes would be 11. So, the time is 1:11. The hour 1 is valid. The minutes 11 are valid (less than 60). So, 1:11 is a single digit repeating time.

Let us check other single digit hours: If the hour is 2, minutes are 22. So, 2:22. This is valid. If the hour is 3, minutes are 33. So, 3:33. This is valid. If the hour is 4, minutes are 44. So, 4:44. This is valid. If the hour is 5, minutes are 55. So, 5:55. This is valid.

If the hour is 6, minutes would be 66. But minutes cannot be 66. Minutes must be less than 60. So, 6:66 is not a valid time. Any hour greater than 5 will make the minutes 66 or more. These times are not possible.

The single digit repeating times are:

$\boxed{\text{1:11, 2:22, 3:33, 4:44, 5:55}}$

Diagram 1

Step 2 — Palindromic times

A palindromic number reads the same forwards and backwards. For times, we look at the digits of the hour and the digits of the minutes. There are two main patterns for palindromic times on a 12-hour clock.

Let us check hours from 1 to 12.

Part A: For single-digit hours (from 1 to 9)

For these hours, the time is palindromic if the minutes start with a 0. The minutes must end with the same digit as the hour. So, if the hour is H, the minutes are 0H.

Let us check for hours from 1 to 5: If the hour is 1, the minutes are 01. The time is 1:01. The digits are 1, 0, 1. This reads the same forwards and backwards. The minutes 01 are valid. So, 1:01 is a palindromic time.

If the hour is 2, the minutes are 02. The time is 2:02. The digits are 2, 0, 2. This reads the same forwards and backwards. The minutes 02 are valid. So, 2:02 is a palindromic time.

If the hour is 3, the minutes are 03. The time is 3:03. The digits are 3, 0, 3. This reads the same forwards and backwards. The minutes 03 are valid. So, 3:03 is a palindromic time.

If the hour is 4, the minutes are 04. The time is 4:04. The digits are 4, 0, 4. This reads the same forwards and backwards. The minutes 04 are valid. So, 4:04 is a palindromic time.

If the hour is 5, the minutes are 05. The time is 5:05. The digits are 5, 0, 5. This reads the same forwards and backwards. The minutes 05 are valid. So, 5:05 is a palindromic time.

Part B: For two-digit hours (from 10 to 12)

For these hours, the time is palindromic if the minutes are the reverse of the hour number. Let the hour be H1H2. The minutes will be H2H1.

If the hour is 10, the digits are 1 and 0. The reverse of 10 is 01. The minutes are 01. The time is 10:01. The digits are 1, 0, 0, 1. This reads the same forwards and backwards. The minutes 01 are valid. So, 10:01 is a palindromic time.

If the hour is 11, the digits are 1 and 1. The reverse of 11 is 11. The minutes are 11. The time is 11:11. The digits are 1, 1, 1, 1. This reads the same forwards and backwards. The minutes 11 are valid. So, 11:11 is a palindromic time.

If the hour is 12, the digits are 1 and 2. The reverse of 12 is 21. The minutes are 21. The time is 12:21. The digits are 1, 2, 2, 1. This reads the same forwards and backwards. The minutes 21 are valid. So, 12:21 is a palindromic time.

Combining both parts, the palindromic times are:

$\boxed{\text{1:01, 2:02, 3:03, 4:04, 5:05, 10:01, 11:11, 12:21}}$

Diagram 2

Answer

(i) Single digit repeating times: 1:11, 2:22, 3:33, 4:44, 5:55 (ii) Palindromic times: 1:01, 2:02, 3:03, 4:04, 5:05, 10:01, 11:11, 12:21

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