jmc

Question

A palindrome is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour value. Therefore, 01:10 may not be used.)

Accepted Answer

First, consider the three-digit palindromes. There are

9

choices for the first digit (the hour):

1

,

2

,...,

9

. There are

6

choices for the second digit (the tens digit of the minutes):

0

,

1

, ...,

5

. The last digit (the units digit of the minutes) has to be the same as the first digit. So there are

9 \cdot 6 = 54

three-digit palindromes.

Second, consider the four-digit palindromes. The first digit (the tens digit of the hour) must be

1

. There are

3

choices for the second digit (the units digit of the hour):

0

,

1

, and

2

. The third digit must be the same as the second digit, and the fourth digit must be the same as the first digit. So there are

3

four-digit palindromes.

In total, there are

54 + 3 = 57

different palindromes.