Midnight on Which Day?

Date: 04/12/2003 at 19:27:22
From: Daniel Cash
Subject: What is the deal with midnight

On a clock, the second hand only moves 6 degrees every 7 seconds, or 
seven times as slow. If the hour and minute and second hand are all 
pointing at 12 at midnight on Sunday, at what day and time will they 
next be in this position?

What exactly does midnight mean?

This question is pretty easy and straightforward (GCF of 12 and 7 
is 84 hours or 3.5 days).  But the problem occurs when reporting my 
answer. I assumed midnight on Sunday meant Sunday at 12:00 AM. So 
three and half days later would be Wednesday at 12:00 PM or noon. When 
I answered this at a math competition today, I was told it was 
Thursday at noon because midnight on Sunday implied it was almost 
Monday. I understand how this can be interpreted in that way. My 
question is "Is there an appropriate way to read this?" and "Is the 
second interpertation just a commonly accepted way of reading this?"

Date: 04/12/2003 at 20:34:45
From: Doctor Rick
Subject: Re: What is the deal with midnight

Hi, Daniel.

I believe the question was unfairly ambiguous. Look at this page from 
the National Institute of Standards and Technology - an authority on 
time if there is one:

   http://www.boulder.nist.gov/timefreq/general/misc.htm 

It does not answer your question exactly, but rather the question of 
whether noon or midnight is AM or PM. You assume midnight is 12 AM, 
but it is not; it is neither. But note what the site says:

"When a specific date is important, and when we can use a 24-hour 
clock, we prefer to designate that moment not as 1200 midnight, but 
rather as 0000 if we are referring to the beginning of a given day 
(or date), or 2400 if we are designating the end of a given day (or 
date).

"To be certain of avoiding ambiguity (while still using a 12-hour 
clock), specify an event as beginning at 1201 a.m. or ending at 1159 
p.m., for example; this method is used by the railroads and airlines 
for schedules, and is often found on legal papers such as contracts 
and insurance policies."

Thus the advice is NOT to refer to midnight on a certain day; this 
phrase is inherently ambiguous. 

I would add that in certain contexts there is no ambiguity; for 
instance, "Tax forms must be postmarked by midnight on April 15." (I 
found a similar phrase on the NIST site itself.) It's clear from the 
word "by" that we are thinking of midnight as the end of a period, so 
we have a reasonable expectation that midnight here is seen as the 
end of a day. But I do not see such a reasonable expectation in the 
context of your problem. If anything, the context refers to midnight 
as the *beginning* of a period, so there is reason to see midnight as 
the beginning of Sunday, as you did. 

The problem is poorly stated and either interpretation should be 
accepted.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/