I was chatting with an older friend of mine when I happened to mention the ages of my two children. He said to me, "when your youngest was born your age was the square of the older one. Now your age is the product of both their ages."

I then thought about his age and the ages of his children and replied, "When your youngest was born your age was the product of the older two. Later on your age was the product of the oldest and youngest. Now your age is the product of the younger two."

What are our two ages?

(Assume we all have the same birthday.)

Let the respective ages of the man, his youngest child and the oldest child be A, B and C.

Then, by the problem:

(i) A=B*C with C > B
                      2  
(ii) A-B = (C-B)  
                                                                          2
Let C-B=x, then from (ii) we must have: A-B = x  so that from (i), we have:

      2
B+x  = B(B+x)
                           2                  
                         x
or,   B+x = 1 + --- ...... (iii)
                         B
                 
                                            
If x=B, then, (iii) => 2B = B+1=> B = 1. Hence, x=1, so that: C=2, giving: A=1*2=2, which is inadmissible, since no father's age can be 2.


If, x< B then, B+x < 1+B, so x < 1. Accordingly, x=0, which yields A=B=C, which is a contradiction.

Consequently, we must have x > B ....(iv)

Then,
 
 2           2                   2                       2
x < B + x  = B(B+x) = B +Bx < (B + x/2)

or, x < B + x/2

or, B > x/2  ..... (v)
                                                               
                                                                       2
Combining (iii), (iv) and (v) we observe that B is a factor of x  with the restriction x/2 < B < x ----(vi)

                                                                               
The normal lifetime of a human being is at most 100 years.
                                    2
So, 100 >= A =B*C > B  (since C>B )=> B <10=> B<=9 ....(vii)

                                                                              2
If x is a prime number, then there is no factor of x  satisfying the restriction in (vi) unless x/2=1 or, x=2
which again violates the said restriction.
                                                     a
If x is a prime power, so that x= p where p is a prime number, then we must have x/2 > x/a, so that a<2, or a=1,
which is not a prime number and hence a contradiction.

The only positive integers less than or equal to 9 which are not a prime or a prime power are 1 and 6 so that x= 1, 6.
It has been shown earlier that x=1 is not valid.

For x=6, the only  factor of 36 which is greater than 3 but less than 6 is 4. Accordingly, B=4.

Hence, C=4+6=10, giving: A = 4*10=40

Consequently, the respective ages of the man, his youngest child and the oldest child are 40, 10 and  4.

Edited on January 22, 2013, 2:36 pm

Edited on January 22, 2013, 2:47 pm

Edited on January 22, 2013, 3:16 pm

Posted by K Sengupta on 2013-01-22 14:25:15