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# Hand-To-Hand Switcheroo

The Puzzle Corner activity this week is a magic trick that requires no slight of hand, just a little dexterity of hand—and an application of topological principles. In the trick, two small cylindrical objects are switched back and forth between hands without dropping them. This is not as easy as it might seem since the objects must start off being held as shown in the first illustration on the student sheet and end up being held as shown in the second illustration. There are two challenges in this Puzzle Corner activity: learning how to do the trick, and then explaining how it works.

Our neighbor introduced us to this trick over ten years ago. One day when we were at his house, he placed a small spool of thread in each hand so that the spools were held in the crooks between his thumbs and index fingers. He grasped the ends of the spool in his left hand with the thumb and index finger of his right hand, and vice versa. He did this in such a way that he could quickly pull his hands apart, and be left holding the spools between his fingers without dropping them or letting go of them at any time. He then handed us the spools and challenged us to do the trick. Each time we tried, the spools and our fingers formed two interlocking circles like the links of a chain and we couldn’t get them apart. We thanked our neighbor for providing us with a challenge and went home to try and solve this intriguing puzzle on our own. After much persistence on both of our parts, we were able to replicate the trick. (Michelle solved it first!) Once we had figured out how to do the trick, we practiced it until we could make the switch as easily as our neighbor. At this point, we were able to carefully examine how this switch was possible and see how it related to the mathematical field of topology. This trick is now one of our favorite things to show students and adults alike, and so we are sharing it with you in this column.

To do Hand-to-Hand Switcheroo, you’ll need to find some objects to switch from hand to hand. Since small spools of thread are not easy to find, especially in a classroom setting, you can substitute other small cylindrical objects like pieces of chalk, crayons, AA batteries, etc. These objects should be about five centimeters (two inches) long and at least half a centimeter in diameter.

Before you do this activity with students, you should master the trick yourself. Doing this may take some persistence and practice—two important characteristics of a good puzzle solver. Once you’ve mastered the trick, you can show it to your students just as our neighbor showed it to us and then step back and enjoy watching your students’ efforts.

To start, get two pieces of chalk, crayon, etc. and place them in your hands as pictured in the first diagram on the student sheet. Grab each object by the end with your thumb and middle finger, and try to switch the hand in which they are held without dropping them or letting go of them. If you are successful, the pieces will be held as shown in the second diagram on the student sheet. Please exercise persistence and try to perform this trick successfully. Once you have figured out how to do it, practice the trick until you can make the switch smoothly without thinking about what you are doing. At this point, you are ready to introduce it to your students and think about why it works.

Show students the switch several times, and then give them the objects you have selected and let them try it for themselves. It is unlikely that your students will succeed in making the switch after only a few tries, so encourage them to be persistent. Show your students the switch in slow motion as many times as they request—after all, you want them to be able to perform the trick. When one or two students are successful, encourage them to show others how to do the trick. Once students have mastered the trick they can try to explain how it works and challenge their friends and family with it!

To do this trick you need two pieces of chalk or crayon about 5 cm long. Place the pieces in your hands as shown below.

Then, without dropping them or letting go of them at any time, make the pieces switch hands. At the end of the switch, you should be holding them as shown in the drawing below.

How does this trick work?
What did you have to do to switch the pieces?

Solution

Click the arrow below to view the solution.

1. Begin with two objects held in the crook between your thumb and middle finder, as shown.

2. Using your right hand, grasp the object in your left hand with the thumb and middle finder, as illustrated.

3. Twist the left hand so that the thumb grabs the end of the object in your right hand that is closest to your palm, and your middle finder grabs the other end of the object. Your middle finger from your left hand must go on the outside of the thumb on your right hand.

4. Pull your hands apart. If you have done it correctly, the objects will now be held between the thumb and middle fingers of the opposite hands. One of the objects will be upside down with respect to its starting position.