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| A Birthday Problem (Posted on 2007-02-08) |
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When Mr. Grimsby asked Mr. Fleming his age on a certain day in the current century, Mr. Fleming responded cryptically, "Today's day of the month corresponds to my age in years. My birthday will occur before this week is over, and then, the day of the month will precisely correspond to one fifth of my new age."
When is the most recent date in which this could have happened?
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Submitted by K Sengupta
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Rating: 2.5000 (2 votes)
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Solution:
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(Hide)
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TomM has posted an analytical solution in this location.
I solved this problem in the following manner:
By the conditions of the problem, during the course of the current week, Mr. Fleming will be precisely 5x years old on xth day of the next month.
Accordingly, Mr. Fleming is (5x-1) years old on "today" which must correspond to the (5x-1)th day of the current month.
Let the total number of days in the current month be d.
Accordingly, in terms of provisions governing the conditions of the problem;
d+x- (5x-1)<7 ...(i)
Since, any given month must necessarily consist of at least 28 days, it follows that d>=28.
Hence, from (i) we obtain:
28+x - (5x-1)<7
Or, 4x>22
Or, x> 6 + 1/2
Since, any date of a given month is an integer, it follows that, x must be an integer.
Therefore, from (i), we obtain:
x>=6.
However if x>=7, then (5x-1)>=34; which is a contradiction since no month in any given year can consist of 34 or more days.
Consequently, x=6; giving:
5x-1 = 29, so that x= 6
Therefore, Mr. Fleming is 29 years old on today which is 29 February and will be 30 years old on March 6.
Since, it has been conclusively demonstrated that the conversation took place on a leap year, that is, an year having 29 February as a valid date, and since the problem was posted on 2007-02-08, it follows that:
The most recent date in which the statement made by Mr. Fleming is possible must be Sunday February 29, 2004.
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Also refer to Charlie’s post for other probable dates for occurrence of the said event in this location.
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