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### The Vicar and the Curate (Ages of Three People)

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Date: 03/20/2002 at 23:20:40
From: Jason Mooney
Subject: Crazy problem

This is a problem that I just can't figure out. Please help.

Note: Vicar and Curate are both terms for clergypersons (ministers,
pastors, priests).  These words are more commonly used in England
than in the USA.

Problem: THE VICAR AND THE CURATE

A vicar and a curate, both with highly developed mathematical
intelligence, were walking along a road, when they saw three people
coming toward them.
"Here is a little problem for you, curate," said the vicar. "The
product of the ages of those three people is 2450 and the sum is twice
your age. How old are those people?"
The curate thought for a while and then said, "I cannot answer your
question."
The vicar, in turn, thought for a while and then said, "I see now.
You are quite right, of course. Well, here is another piece of
information, which will allow you to work the problem. I am older
than any of those three people."

HOW OLD IS THE VICAR?
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Date: 03/21/2002 at 01:33:47
From: Doctor Jeremiah
Subject: Re: Crazy problem

Hi Jason,

The prime factorization of 2450 is: 2x5x5x7x7

That means the people could be only one of these sets of ages:

2,   5, 5x7x7  ->   2,  5, 245  ->  sum = 252
2,   7, 5x5x7  ->   2,  7, 175  ->  sum = 184
2, 5x5,   7x7  ->   2, 25,  49  ->  sum =  76
2, 5x7,   5x7  ->   2, 35,  35  ->  sum =  72
2x5, 5x7,     7  ->  10, 35,   7  ->  sum =  52
2x5,   5,   7x7  ->  10,  5,  49  ->  sum =  64
2x7,   5,   5x7  ->  14,  5,  35  ->  sum =  54
2x7, 5x5,     7  ->  14, 25,   7  ->  sum =  46
2x5x7,   5,     7  ->  70,  5,   7  ->  sum =  82
2x7x7,   5,     5  ->  98,  5,   5  ->  sum = 108
2x5x5,   7,     7  ->  50,  7,   7  ->  sum =  64

Now the curate also knows that the sum is twice his age. He certainly
knows his age and if one set of ages were uniquely twice his age he
would have it figured out, so the only way he wouldn't have it figured
out is if it were one of the two sets with the same age.

2x5,   5,   7x7  ->  10,  5,  49  ->  sum =  64
2x5x5,   7,     7  ->  50,  7,   7  ->  sum =  64

Both of these add to 64 so he can't know which of them is the right
one (if it were any other it would be obvious that they were the only
ones twice his age).

Given the last clue, that the vicar is older than all three people,
there are three possibilities for the vicar's age:

1. The vicar is younger than 50.
2. The vicar is 50.
3. The vicar is older than 50.

In case 1, neither of the solutions would work. So the vicar can't be
younger than 50.

In case 3, his age doesn't resolve the ambiguity. So he can't be older
than 50.

That leaves case 2. The vicar must be 50. And if the vicar is 50, only
one of the solutions will satisfy the constraint.

2x5,   5,   7x7  ->  10,  5,  49  ->  sum =  64

Note that this ignores cases where one of the people is only a year old!  But that doesn't really change the outcome of the problem, since they lead to unique  sums.

- Doctors Jeremiah and Ian, The Math Forum
http://mathforum.org/dr.math/
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