102. A Knotty Problem 2012-5-28

After a day of good interviews, Eddie's prospective supervisor, Sam, invited Eddie to join Sam and half a dozen prospective colleagues for a drink after work.

At the bar, they nurse their drinks and engage a series of complete strangers in intriguing bar bets, always winning the money, but never making friends.

Eddie sees that he has 90 minutes to board his flight home. "Sam, everyone, thanks for the excellent day of interviews. You had good questions, and helpful answers to mine. I need to catch a cab to the airport. I hope we'll speak soon about the next step in this process."

Sam nodded and said, "You're tied with another candidate, except that he didn't pass one test that we haven't given you, yet. It'll only take a minute. Would you like to try?"

Eddie thinks, I don't really have a choice. He says, "Let's do it!"

Sam reaches into his jacket pocket, pulls out three feet of what looks to be clothesline, and lays it out on the bar, so it looks like a line segment from left to right. "Eddie, listen carefully. I want you to grab one end of that rope with your left hand, and without letting go, grab the other end with your right hand, and—never letting go—tie a knot in the rope. You have until the taxi comes."

Sam turns to an underling and says, "Flag down a cab for Eddie, please. He's in a hurry."

Eddie looks at the rope and thinks, That's impossible.

Is it? If not, what should Eddie do?

Answer

Eddie should hold his left arm straight in front of him at shoulder height and horizontal. He should do the same with his right arm, bend his right arm 90 degrees to the left at the elbow, and let his right wrist rest on his left bicep. Next, bend the left arm back toward the body and let the left hand go under his right bicep.

Holding that position, he should bend at the waist, awkwardly grab one end of the rope with one hand, then awkwardly grab the other end with the other hand.

Finally, holding firmly to the ends of the rope, he should separate his hands, until the rope is taught—with a an overhand knot in it.

This problem, an ancient chestnut, appears in Math Trek: Adventures in the Math Zone, p. 5. For more information, see "Reference Desk", this page, column 3.