I own a very unique clock. It is digital, but it displays the time in Roman numerals. It does this in the same way as a normal digital clock (hours : minutes). The clock does not display seconds. For example, at 7:34 the clock displays "VII:XXXIV". Whenever a zero is required it is represented by a dash (-). That is, 1:00 is represented by "I:-".

Sometimes I place this clock on a flat, horizontal mirror which reflects what is displayed on the clock.

For what proportion of the day does the reflection of the time on the clock look the same as the time displayed by the clock itself?

Please provide an answer for both 12hr timing (the clock does not indicate the difference between AM and PM) and 24hr timing.

First a few assumptions:
1.  That 4 is written IV, rather than IIII  (sometimes seen on clocks, and I believe an older form of Roman Numerals)
2.  That 40 is written XL

In a 12 hour clock, 7/12 of the hours would reflect accurately, and in a 24 hour clock 13/24 of the hours would.  For any hour that reflects accurately, 4 of the 6 tens column of minutes are OK, and 5 of the 10 ones column.  So for any OK hour, 2/3 * 1/2 = 1/3 of the minutes are "OK".

12 hour clock:  7/36
24 hour clock: 13/72

Posted by Larry on 2004-05-16 10:58:29