As corn mazes go, the one on Bob Connor’s farm in Danvers, Mass., isn’t particularly challenging.

“It’s one of our average-to-smaller mazes,” said Brett Herbst, who counts that maze among the 2,000 he has designed in the past 16 years. A typical visitor should expect to complete it in about 20 minutes, he said.

Still, a local family had to be retrieved by the police on Monday when they were unable to find the exit to Mr. Connor’s maze before the sun set. The parents, toting an infant and a small child, panicked and called 911, setting off a chain of events that soon turned them into a target for late-night jabs from television hosts like Jay Leno and Chelsea Handler. (The punch line, Mr. Connor said, was that the family was about 25 feet from the exit when they called for help.)

Modern corn mazes are complex systems designed with the aid of sophisticated computer programs. But they are meant to be challenging, not life-threatening, according to maze builders, and getting out should never require a police escort.

“The great thing about designing mazes is it’s like playing a game of chess” against the customer, said Adrian Fisher, a British designer who holds the world record for the longest maze path ever built. “You plan all your moves in advance, yet in the end you’ve got to be willing to lose.”