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 A prime time at KP (Posted on 2006-05-31)

Military time is based upon a 24-hour clock. The time is expressed in the form abcd hours where a, b, c, and d are base 10 digits, ab is the hours, ranging from 00 to 24, and cd is the minutes, ranging from 00 to 59. (0000 and 2400 hours are equivalent, but one minute past midnight is still represented as 0001).

From his unit, Buck Private Kilroy was assigned to K.P. where he reported to duty on the hour. After a quick, but complete, demonstration on how to peel a potato by the mess sergeant, Kilroy was ordered to peel two potatoes every minute from abcd hours until efgh hours.

Upon timely completion of the task, the buck private was put to work in other kitchen duties for the remainder of the detail (where he could contemplate the meaning of "hurry up and wait") while waiting for the other members of his company to be called to chow hall at (hgfe + abcd) hours.

The following is known:
(a) The times, abcd, efgh, and (hgfe + abcd), are prime numbers;
(b) Each two digit hour and minute of efgh and (hgfe + abcd) are also prime numbers; and,
(c) The sum of the digits of abcd is divisible by three.
(d) Normal addition rules apply in adding the numbers representing "time" values. If a sum results in a value that is not in the form of a valid time, it obviously does not qualify as a solution.

According to the information given in the problem above, answer the following:
(1) What hour did Kilroy finish peeling potatoes?
(2) Given that the number is prime, how many potatoes were peeled?
(3) And, what time was the company of soldiers called to mess?

 See The Solution Submitted by Dej Mar Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution | Comment 1 of 3

Given that abcd is prime, yet the sum of its digits is divisible by 3, abcd can only be 0003.

For hgfe + abcd to be prime, e must be either 0 or 2.  h must be greater than or equal to e, so if e is 2 then h must also be 2.  However, h cannot be 2 as then gh would not be prime, so e = 0 and h must be equal to 1.

So far we have:

abcd = 0003
efgh = 0--1
hgfe + abcd = 1--3

To satisfy the conditions that the two digit hours and minutes of efgh and (hgfe + abcd) are all prime, then possible values of f are 2 and 5, and possible values of g are 1 and 3.

This leaves only four possibilities:

`efgh   (hgfe + abcd)--------------------0211       11230231       13230511       11530531       1353`

Of those four pairs, only the first pair are both prime.  Therefore:

abcd = 0003
efgh = 0211
hgfe + abcd = 1123

(1) 0211 hours.
(2) If he peeled 2 potatoes every minute from 0003 hours until 0211 hours, he peeled 416 potatoes.
(3) 1123 hours.

Edit: Haha, whoops.  I forgot to get out of base 10 mindset.  If he peeled from 0003 hours to 0211 hours, that is 128 minutes, not 208.  He peeled 256 potatoes.

Or, since the question says "Given that the number is prime", perhaps the intended answer is 257 potatoes - 256 peeled by Kilroy plus the one peeled in the demonstration.

Edited on May 31, 2006, 10:03 am
 Posted by tomarken on 2006-05-31 09:57:30

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