Question

A palindrome is a whole number that reads the same forwards as backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are 1:01, 4:44, and 12:21. How many times during a $12$ -hour period will be palindromes?

$57$

$60$

$63$

$90$

$93$

Solution:

For a single-digit hour $h$ (from $1$ to $9$ ), the display $h\!:\!m_1 m_2$ reads the same backwards when $m_2 = h.$ The tens digit $m_1$ can be $0,$ $1,$ $2,$ $3,$ $4,$ or $5,$ giving $6$ palindromes per hour, so $9 \times 6 = 54.$

For the two-digit hours $10,$ $11,$ $12,$ the display $h_1 h_2\!:\!m_1 m_2$ is a palindrome when $m_1 = h_2$ and $m_2 = h_1,$ giving exactly 10:01, 11:11, and 12:21 — one each, for $3$ more.

The total is $54 + 3 = 57.$

Thus, the correct answer is A .

1988 AMC 8 Problem 25

Solution:

Problem 25 in Other Years