When Grandpa was over fifty but less than eighty, he told to a friend: "each of my sons has as many sons as...

arithmetic/grandpa.sons New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

When Grandpa was over fifty but less than eighty, he told to a friend:
"each of my sons has as many sons as brothers, and the number of my
years is exactly that of the number of all my sons and grandsons."
How many sons and grandsons had Grandpa?

-- Rocky

Show solution

Let "s" represent the number of sons grandpa has.
Let "g" represent grandpa's age.

If each of grandpa's sons has as many brothers as he has sons, then each
of them has (s - 1) sons. Therefore, grandpa has a total of s(s - 1)
grandsons. If the total number of male family members below grandpa
equal his age, then:

Grandpa's age = [Number of sons] + [Number of grandsons]
g = s + s(s-1)
  = s + s^2 - s
  = s^2

Since grandpa's age is an integer, and the only square number between 50
and 80 is 64, grandpa is 64. Therefore, s = 8. We conclude that grandpa
has 8 sons and 56 grandsons.

-- Santiago Barona <santiago@stg.brown.edu>