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| A Prime Time And Square Time Problem (Posted on 2006-03-16) |
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A meeting started after 1 o'clock but before 12 o'clock according to a wallclock( which does not have separate second marks and minute marks) mounted on a wall of the meeting room. At the precise instant of commencement of the meeting, the second hand ( the clock hand which measures seconds) was situated exactly on a minute mark and a prime number of minute marks ahead of the hour hand.
The 24 Hour electronic wristwatch, belonging to the Chairman who had synchronised his watch in accordance with the wallclock, displayed precisely AB:CD Hours( in HH:MI format ) at the same instant, where the number ABCD in the decimal system corresponds to a perfect square. (ABCD may or may not contain leading zeroes).
Determine the precise time of commencement of the meeting in accordance with the Chairman's wristwatch.
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Submitted by K Sengupta
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Rating: 3.6667 (3 votes)
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Solution:
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The precise time in terms of the Chairman's wristwatch at the time of commencement of the meeting was 03:24 hours.
EXPLANATION:
According to the wallclock, let us assume that the meeting commenced precisely at 12R minutes Q o'clock where 0<=R<=4 and 1 < Q < 12.
Let the second hand be situated precisely S minute marks ahead of the hour hand.
By the problem S is a prime number.
Since, the hour hand is exactly on a minute mark, it follows that the second hand must be on 60 minute mark giving P+S = 60 or, S = 60 - P ; where P= 5*Q + R. ( Refer to my methodologies in deriving a Generalised Solution to "What time is it again?" problem; Location: http://perplexus.info/show.php?pid=3211&cid=24709 and
http://perplexus.info/show.php?pid=3211&cid=25397).
Also, by conditions of the problem:
# ABCD is a perfect square.
# 01< AB< 23 with AB not equal to 12,13 and 00 < = CD < = 59
# AB = Q or Q+12 and CD = 12R with 0 < = R < = 4 and 1 < Q < 12
# S=60-P , with P = 5*Q +R.
Now, ABCD is a perfect square satisfying the other three conditions, whenever:
ABCD = 0324, 0400, 0900, 1024, 1600 and 1936.
Checking each of the abovementioned six values of ABCD, we observe that S is a prime number only when, ABCD = 0324, since ABCD = 0324 yields Q=3, R=2 so that, P =17 giving, S= 60-17 = 43, which is a prime number.
S is not a prime number corresponding to the other five choices of ABCD.
Consequently, the precise time in terms of the Chairman's watch at the time of commencement of the meeting was 03:24 Hours.
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