The idea was simple: David and Sarah Altman planned to buy a large lot with a deteriorating house, tear it down and build something new. Something modern.

As is often the case, however, money got in the way. Building a new house turned out to be prohibitively expensive, so the Altmans settled for the next best thing: remodeling something old. And when you live in Portland, Ore., as they do, and you’re looking for an old house, there are lots to choose from.

In the residential neighborhoods surrounding downtown, you’ll see many historic architectural styles — Craftsman, Victorian, Queen Anne — but perhaps none more plentiful than the American Foursquare, named for the four large rooms typically found on each floor. With its plain lines, the style was a reaction to the excesses of late-19th-century architecture and was popular here in the early decades of the 20th century. For a couple enamored of modernism, it seemed like a good choice.

The 1910 Foursquare house in the Richmond neighborhood that the Altmans bought in 2013, for $420,000, had three bedrooms and 2,400 square feet of interior space. Their plan was to do “a light remodel,” said Ms. Altman, 48, who owns a corporate executive sales training company. They initially intended to sell it to fund a brand-new house of their dreams, but they later reconsidered.