Explore the Solution Space. Now, unless the sum of the ages is 13, the census-taker would not have been stumped. However, if the sum is in fact 13, then he would still be left with two possibilities: the triples (2, 2, 9) and (6, 6, 1). Where do we go from here? How could the mother be so certain that knowing the sum of the ages would not be enough information for the census-taker? It must be that the three ages add up to 13. The only clue that we have not yet used is the fact that the oldest daughter Annie likes dogs. But what does that have to do with anything? Well, of the two possible combinations (2, 2, 9) and (6, 6, 1), only one of them represents having an “oldest daughter”. Therefore the ages of the three daughters are 2, 2, and 9.
(Your Turn: A perfect polygon is a polygon with integer sides for which area equals perimeter. Find all perfect rectangles.)