The question as it appears in the 02/27 edition of Specs & Techs from GlobalSpec:
Joe was thinking about his grandfather's age; he found that if he added up all the birthdays he has had, including his age now, the result was one year more than his grandfather's age. Joe also discovered that if he added the two digits of his grandfather's age, the result was his own age. What ages are Joe and his grandfather?
(Update 8:30 AM EST 03/06/07) And the Answer is....
Let x be Joe's age and g his grandfather's age.
We can express g = 10a + b, where a is the most significant digit and b the least significant digit. Now we have two equations
x = a + b (1)
x(x+1)/2 = 10a +b +1 (2)
Substituting the first equation into the second and expanding we get
x2  x = 18a + 2
Complete the square of the lefthand side and simplify to get
(2x  1)2 = 9(8a + 1)
or
2x  1 = 3 sqrt(8a + 1) (3)
also we know that a < 10.
Now, to satisfy equation (3) 8a + 1 has to be a perfect square, because a, b, and x are integers. With these conditions there are only 3 values of a that can satisfy the equation: 1, 3, and 6. Substituting these numbers into equation (3) we get three possibilities:
(1) If a = 1, then x = 5 and b = 4. Then Joe's age is 5 years and his grandfather is 14.
(2) If a = 3, then x = 8 and b = 5. Then Joe's age is 8 years and his grandfather is 35.
(3) If a = 6, then x = 11 and b = 5. Then Joe's age is 11 years and his grandfather is 65.
The ages are 11 and 65 years.

Re: Grandfather's Age: Newsletter Challenge (02/27/07)