Henry Ernest Dudeney/Puzzles and Curious Problems/47 - The Quarrelsome Children/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $47$

A man married a widow, and they each already had children.

Ten years later there was a pitched battle engaging the present family of $12$ children.

The mother ran to the father and cried,

"Come at once! Your children and my children are fighting our children!"

As the parents now had each nine children of their own, how many were born during the ten years?

$6$

Let $A$ and $B$ denote the sets of the children of the man and the widow respectively.

Then

$A \cap B$ denotes the set of the children of both of them

$A \cup B$ denotes the set of all the children together.

$\card {A \cup B} = \card A + \card B - \card {A \cap B}$

where $\card X$ denotes the cardinality of a set $X$.

We have been given that:

$\card {A \cup B} = 12$

$\card A = \card B = 9$

which leads to:

$\card {A \cap B} = 9 + 9 - 12 = 6$

$\blacksquare$