From Ryan Nelson comes a puzzle related to the digital display of digits (for the second week in a row!):
A digital 12-hour clock displays 10 digits: two digits representing the hour (from “00” to “12”), two digits representing the minute, two digits representing the second and four digits representing the year.
When will the clock next use every digit from 0 to 9?
Consider the format, HH:MM:SS YYYY, and the following table to summarize the possible placements.
| Place | Digit Possibilities | Comment |
|---|---|---|
| H1 | $1$ | Only option. |
| H2 | $0-2$ | Must be in order to use all 10 digits. |
| M1 | $0-5$ | Must be less than 6. |
| M2 | $0-9$ | Any. |
| S1 | $0-5$ | Must be less than 6. |
| S2 | $0-9$ | Any. |
| Y1 | $2$ | Assume 2000s, for now. |
| Y2 | $0-9$ | Any. |
| Y3 | $0-9$ | Any. |
| Y4 | $0-9$ | Any. |
$3-5$ remain for the tens place in minutes and seconds. $3$ should be the hundreds place in year, leaving $4$ for minutes and $5$ for seconds.
$6,7$ should be the remaining year digits, and $8,9$ should be the remaining ones for minutes and seconds, respectively.
10:48:59 2367 is the next time all digits will be used.