How many palindrome times on a 24-hour clock?

Question

A 24-hour digital clock shows the time as four digits, HH:MM (from 00:00 to 23:59). A time is a palindrome if the four digits read the same forwards and backwards, for example 02:20 (digits 0, 2, 2, 0) or 13:31.

How many palindrome times occur in a single day?

Accepted Answer

There are exactly 16 palindrome times per day. Set up the condition. Write the display as digits d1 d2 : d3 d4. For a palindrome, the first digit must equal the last, and the second must equal the third: d1 = d4 \quad	ext{and}\quad d2 = d3. So the minutes are completely determined by the hour: the minute digits are the hour digits reversed (d3 = d2, d4 = d1). Enforce a valid minute. The minute's tens digit is d3 = d2, which must be 0–5 (minutes only go up to 59). So we need the hour's ones digit d2 to be at most 5. (The minute's ones digit is d4 = d1, always fine.) Count valid hours 00–23 wh…

How many palindrome times on a 24-hour clock?

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