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Perplexing Pencil Problem

Perplexing Pencil Problem

This week’s post introduces a wonderful topological puzzle. Topology is one of the newest fields in mathematics. To illustrate this, note that Henri Poincare’ (1854-1912), who is considered the founder of algebraic topology, published the first systematic treatment of topology in 1895. On the other hand, Euclid (330?-275? BCE), the father of geometry, wrote his 13-volume work on geometry, Elements, more than two thousand years earlier. As you can see, topology is a relative newcomer in the field of mathematics.

Topology, which is closely related to geometry, is concerned only with those properties of geometric objects (such as number of holes, dimensionality, and boundaries) that remain unchanged when the objects are distorted in any way by such things as twisting, shrinking, or stretching. Because of this, topology has sometimes been called “rubber sheet geometry.”

The Perplexing Pencil Puzzle utilizes this rubber sheet geometry to create a seemingly simple, yet for most people, extremely challenging puzzle. In this puzzle a pencil with a string loop is attached to a buttonhole while the person who will be challenged to remove it watches. It is a rare person who can remove the pencil and loop, even after seeing it being attached.

The Perplexing Pencil Puzzle presented here is a homemade version of Sam Loyd’s (1841-1911) commercially produced Buttonhole Puzzle. Loyd, perhaps the premiere puzzlist of all times, invented this puzzle as a sales gimmick. In their book, Puzzles Old and New: How to Make and Solve Them, Slocum and Botermans relate the puzzle’s historical background: “It was made for the President of the New York Life Insurance Company and designed to help insurance agent’s [sic] sell life insurance. Loyd relates an amusing story. He showed the puzzle to the President, John McCall, who was not impressed. Loyd fixed it to McCall’s buttonhole and bet him a hundred dollars to one that he couldn’t get it off in half an hour without cutting the string. McCall couldn’t and lost his dollar. Loyd then said ‘I’ll take it off for you if you’ll agree to take out a ten thousand dollar policy on your life!’The puzzle became famous and Loyd said it was one of his most successful. The insurance agents must have used it with success ‘to buttonhole’ became an expression meaning to grab somebody’s attention.” (p. 114)

To make this puzzle, you need a pencil and some string. I like to use a new, unsharpened pencil, so students won’t accidentally poke themselves when trying to solve the puzzle. This also keeps pencil marks from getting on the shirt. Drill a hole through the pencil just under the metal band that holds the eraser. The easiest way to do this is with an electric drill using a small bit, but you can also use the point on a compass, ice pick, or other similar object to drill the hole by hand. Thread the string through the hole and tie a knot in it so that the loop created is about one to two centimeters shorter than the pencil.

After making the puzzle, you will need to practice attaching and detaching it before introducing it to your students. Find an old shirt with buttonholes large enough for the pencil to pass through easily. Follow the steps on the activity sheet to attach the pencil and loop of string to the shirt. After doing this, try to remove the pencil without cutting the string or shortening the pencil. (Some enterprising students have solved this puzzle using the pencil sharpener!) If you can think “topologically” you should be able to remove the pencil and solve the puzzle.

There are numerous ways to introduce this puzzle to your students. One way is simply to take the shirt and pencil setup to class and attach the pencil while students watch. After doing this, place the shirt at a center and individuals or small groups of students can work on the puzzle in their free time. If you use centers, several puzzle setups can be placed in a center. A more fun, but potentially risky, way to introduce the puzzle is to make multiple pencil setups and attach them to your student’s clothing. You must use proper discretion if you choose this approach. I only attached the pencils to my male student’s shirt (with hilarious results) or to either male or female student’s coats or jackets, which they weren’t wearing at the time. It is getting harder to use this latter approach since fewer clothes, especially those worn by students to school, have buttonholes nowadays. Use whatever approach best fits you situation, but remember, you will have to solve the puzzle yourself first!

Now that the pencil is attached, all you have to do is remove it and you’ve solved the puzzle. Good luck!



Click the arrow below to view the solution.

To solve The Perplexing Pencil Puzzle all you need to do is to reverse the process you used to attach the pencil and loop to the buttonhole. To do this, loosen the knot so that the loop surrounds the buttonhole. Pull the material around the buttonhole up though the loop so that the material is several centimeters above the loop. Next, poke the eraser end of the pencil through the buttonhole. Once the pencil is through the buttonhole, pull on it from below and it should come out quite easily. If there is not enough slack in the string to free the pencil, pull more material up through the loop and try again.


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