One begins by wrapping the loop (a flexible string about two meters, or about two yards, long) around a few fingers. The player then uses his or her hands, fingers and even teeth to transform the web of string into the predetermined pattern, often telling a story about the figure. In the standard version a second player takes the structure onto his fingers and forms a new one.

Cat's cradle has been played by children throughout the world for thousands of years. Recently two delightful books inspired me to examine the game and some of its variants. One is Spiders' Games: A Book for Beginning Weavers (University of Washington Press, 1979), by Phylis Morrison, who contributes to the "Books" section of this magazine each December. In the book she describes among other things the string figure called "ten men," using it as an example of how one's fingers can serve as a loom [left]. The second book is Orderly Tangles: Cloverleais, Gordian Knots, and Regular Polylinks, by Alan Holden.

The string in each of these games is a loop in a strict topological sense because the figure collapses to a loop when the fingers are removed. According to the common meaning of knot, however, the string figures are knots. I wondered if the procedures for tying these knots could be catalogued. Surely the figures were devised by people who had mentally constructed their own catalogue of procedures. This month I begin a catalogue by analyzing several string figures to be made by a single player.

Analysis requires terminology: I call the string running between two fingers a section and the string passing around a finger or the entire hand a loop. A11 string figures have places where one section crosses another; I call them crossings. In the most intriguing string figures some sections cross each other twice. I call them wraps.

To study the wraps in a string figure I mentally follow along the full length of the string from an arbitrary starting point. When I pass through a wrap, I note the sequence of crossings. If the section I am following (the first section) passes over and then under the second section, I describe the relation as an over-under wrap. If instead the first section passes under and then over the second section, I describe the relation as an under-over wrap.

I began my study with several questions. How is a wrap created? Are wraps governed by rules? For example, must a

string figure have an even number of wraps, as many do? Must it have an equal number of over-under and under-over wraps or can the weaving establish a bias toward one kind? I shall return to these questions after I explain how to begin the game and how to weave the "ten men."

According to W. W. Rouse Ball, whose book is listed in this month's "Bibliography", most string figures are begun with one of three basic openings: A, B and the Navaho. For opening A hold your hands with the palms toward each other, as is shown in Figure 3 (which follows the practice of shading the string sections that are to be rearranged and marking with an X the fingers that are to be moved). Run the string around the back of the little finger, across the palm, around the back of the thumb, across to the other hand, around the back of the thumb, across the palm, around the back of the little finger and across to the starting point on the first hand. With your right index finger reach below the string section on your left palm and pick it up, hooking the string onto the back of the right index finger. Separate your hands to tighten the string.

With your left index finger pick up the string section on your right palm, hooking the string onto the back of the left index finger. Again separate your hands. You now have opening A. Opening B is identical except that you start with the left index finger.

The Navaho opening (named for the Navaho Indians, who were highly skilled at string games) entails a more complicated maneuver. With the string in a figure eight [see Figure 4 ] insert the index fingers in the top loop of the figure and the thumbs in the bottom loop. (Notice that at the crossing point the top section of string runs from your left to your right.) Turn your palms away from you with the fingers upward. This move catches the top loop of the figure eight on your index fingers and the bottom loop on your thumbs. Rotate your hands to make the palms face.

Now for "ten men." Begin with opening A and continue through 14 steps. (1) Seize the far string section with your teeth and pull it over the tops of the other sections. (2) Move your right index finger under the section running from the left little finger to your mouth and pick it up. Keep this captured section near the tip of the index finger to separate it from the string already looped around the finger. (3) Similarly pick up the section running from the right little finger to your mouth. As before, keep this section near the tip of the index finger. (4) Release the string from your mouth and thumbs while you separate your hands to take up the slack.

(5) Bend each thumb away from you so that it passes under four string sections and you can pick up the next section from below (it is the nearer section of the loop around the little finger). Pull the thumbs and the sections they have captured back to you so that the thumbs return to their original orientations. (6) Bend each thumb away from you and pick up from below the nearer section of the string looped around the tip of the index finger. Keep the captured section of string high on each thumb to separate it from the string already looped around the thumb.

(7) With your teeth lift the lower loop at each thumb over the top loop and the thumb tip and drop it in front of the thumb. The procedure is called Navahoing the loops. (8) Release the upper loop on each index finger while you separate your hands far enough to take up the slack. (9) Transfer the loops on each thumb to the top of the adjacent index finger by bending each index finger toward you and then picking up, from below, the loop on the thumb. Keep the captured section at the top of the index finger.

(10) Bend each thumb away from you and under the sections of string on the index finger. Put the back of each thumb under the near section of string on the little finger and pick it up. Return the thumbs to their initial orientation. Keep the loops low on the thumbs. (11) Bend each thumb away from you and pick up from below the nearer string section of the loop around the tip of the index finger. Keep these sections near the top of the thumbs. (12) Navaho the thumb loops(13) Bend each middle finger toward you, passing over two string sections. Pick up the next string section from below. Keep these captured sections near the tops of the middle fingers as you return the fingers to their initial orientations. (14) To display the string figure drop the loops around the little fingers, extend your thumbs, keep the index and middle fingers together, separate your hands and turn the palms away from you.

The "ten men" has several crossings and 12 wraps. Although I mastered the construction in about an hour, I spent many hours in gaining an understanding of how wraps are created. In one method that seems to be common to several string figures I add a second crossing to two string sections that already cross. As the situation is depicted in Figure 5 a crossing lies in front of a finger around which the string is looped. To create a wrap I bend another finger under the sections and pick up the upper section of the crossing from below. The return of the finger to its initial position pulls the captured string section under the lower section of the crossing, forming a wrap.

I can also form a wrap from the initial crossing if I bend a finger over the upper section and pick up the lower section from below. The return of the finger to its initial position pulls the lower section over the upper one, forming a wrap. I call these moves a hook because a finger hooks one of the string sections in order to pull it around the other section. Sometimes the sections that form the initial crossing loop around one finger. Sometimes the sections are looped around different fingers. In each case the hook adds a second crossing to the sections to form a wrap.

In some string figures a wrap is generated by twisting a finger. For example, run a section across your left palm. With your right index finger pick up the string from below. As you separate your hands rotate your finger so that the captured string twists around itself, creating a wrap. When the finger moves through the first half of the rotation, it crosses the string looped around it. In the second half of the rotation the finger forces the lower section to pass over the upper section, thereby creating the wrap.

I have found another procedure that is essentially a hook [Figure 8]. The string loops around the index finger. The near section of the loop crosses under a string section that loops around the thumb. When the loop around the index finger is expanded to loop also around the thumb, a wrap is created.

A double hook is shown in Figure 9. Initially two string sections cross. A third section is pulled under and then over a section on each side of the crossing point, creating two wraps.

The number of wraps that can be produced by Navahoing loops on a finger depends on the shape and orientation of the loops. In Figure 10 the loop that is low on the finger forms a narrower angle than the high loop. Since the loops have the same orientation (their corner angle opens in the same direction), each side of the bottom loop crosses under the corresponding side of the top loop. When you lift the bottom loop over the fingertip and release it, you create two wraps.

In the middle part of the illustration the top loop forms a narrower angle than the bottom loop. Since the loops have the same orientation, the sides of the bottom loop do not cross the corresponding sides of the top loop. This time Navahoing the bottom loop over the top loop does not produce a wrap.

In the bottom part of the illustration only the left side of the lower loop crosses under the left side of the top loop. When the bottom loop is Navahoed, a wrap is created on the left side of the top loop. In sum, the number of wraps created by Navahoing loops on a finger is equal to the number of times the sides of the lower loop crossed under the corresponding sides of the top loop before the Navahoing.

Sometimes making a string figure generates what I call a potential wrap. An example is shown in Figure 11. If the little finger on the right releases the string looped around it and the hands are separated e to take up the slack, the released string wraps around the section that stretches between the index fingers. Many such potential wraps are created during the weaving of the "ten men," but only two of them are converted into real wraps.

Reversing a procedure that creates a wrap eliminates the wrap. In some cases a wrap can also be eliminated by releasing the string from a finger next to the finger you used to Navaho loops. For example, suppose you have just Navahoed the loops on a thumb and one of the loops also passes around the index finger. The wrap between the thumb and the index finger depends on the index-finger loop. If you release e that loop, the wrap disappears.

To sum up, there are three basic ways to produce wraps in the weaving of a string figure: the hook, Navahoing loops on a finger and releasing a loop by a finger to convert a potential wrap into a real one. Included in the hook method is the twisting capture by a finger and the expansion of a loop to include two fingers. Wraps can be eliminated by reversing a procedure or by releasing a loop that stabilizes a wrap you created by Navahoing loops on another finger.

Armed with these basic steps, I dissected the formation of "ten men." In the opening position the string has only 9 crossings. Step 1 creates a potential wrap. (If the string around the right little finger were released, a real wrap would appear.) Step 2 creates another potential wrap. Real wraps do not appear until step 7, where the Navahoing of loops at the thumbs creates a total of four real wraps. Step 8 eliminates the wraps that were between the i2 thumbs and the index fingers because they depended on the loops released by the index fingers in this step. Now there are two real wraps.

In step 10 the thumbs hook string sections to create two new wraps, again giving a total of four real ones. Step 12 creates another four wraps when the thumb loops are Navahoed. I Another two wraps.appear in step 13 when the middle fingers hook string sections. (This hook is subtle. The initial crossing is formed by the captured section and a section that emerges from the far side of the index finger at the top.) Finally, the last two wraps are generated when the little fingers release their loops and convert two potential wraps into real wraps, giving a total of 12 real ones.

Can general rules be devised for the wraps of a string figure? I believe they can. For example, if each manipulation of the string is done on both hands, the number of wraps in the figure must be either zero or an even number. The "ten men" is symmetrical in this way because each move on the right hand is also made on the left hand.

Another rule seems to govern the types of wrap in a string figure resulting from symmetrical procedures. From an arbitrary starting point in the "ten men" follow along the string and note the types of wrap you pass. For example, start at the left thumb and move initially toward the left index finger. As you mentally travel along the entire string, returning to the left thumb, you pass through 12 over-under wraps and 12 under-over wraps. I believe a string figure made by symmetrical procedures will always have as many over-under wraps as under-over wraps.

Trace through the "ten men" again, adding arrows along the way to indicate how you enter each wrap. Since during the full trip you pass through each wrap twice (once along each string section in the wrap), each wrap gets a pair of arrows. They point into a wrap either from the same side of the wrap or from opposite sides. In "ten men" 10 of the wraps have a pair of arrows pointing from the same side. Only the central two wraps have arrows pointing from opposite sides. I believe all string figures produced by symmetrical procedures will turn out to have an even or zero number of wraps where the arrows point from the same side and an even or zero number for which the arrows point from opposite sides.

I have also analyzed the production of wraps in the "fishing net." The figure is initiated from opening A. Release the loops around the thumbs. Bend each thumb away from you and l under three string sections and pick up the farthest string section from below (it is looped around the little finger). Pull the string into place by returning the thumbs to their initial orientation.

You now have one potential wrap formed by the string looped around the left little finger.

Bend each thumb away from you and over one string section. Pick up the next section from below and return the thumbs to their former orientation. This move adds a potential wrap associated with the right little finger. Release the string at each little finger to convert the potential wraps into two real ones. Bend each little finger toward you and over one section Pick up the next section from below and return each finger to its initial orientation. In this move each of your little fingers makes a hook to add another real wrap. Now there are four real wraps.

Release the string from around the thumbs. Bend each thumb away from you and over two string sections. Pick up the next section from below; it is the near side of the loop passing around the little finger. Return the thumbs to their initial orientation. Using your right hand, pick up the loop that passes around the left index finger. Do not remove the loop from the finger but make it larger so that it passes around both the finger and the thumb. This move is essentially a hook because the section you move to the thumb is forced to wrap over another section from the thumb. Now make the same move on the right hand. You have added two real wraps for a total of six.

Navaho the loops on each thumb. Since the loops do not have the same orientation, this move adds only one wrap on each hand, raising the total to eight wraps. Each thumb now has in front of it a small triangle formed by the string section running from the little finger on that hand. Bend each index finger toward you and pass its tip through the triangle on that hand.

To display the figure rotate your hands to put the palms away from you and the fingers upward. Catch on the appropriate index finger the string of the triangle in which it is inserted. (The loop already on the finger slides off.) Release the loops from the little fingers. Catching a string on each index finger adds one wrap for each maneuver. Releasing the string from the little fingers converts two potential wraps into two real ones. Now you are up to 12 real wraps. Between each thumb and index finger is one wrap and an additional crossing. Ten more wraps are spread through the figure.

The "fishing net" follows my improvised rules. The total number of wraps is even because of the symmetry of procedures on the two hands. When I follow along the string from an arbitrary starting point, I find that the number of over-under wraps is equal to the number of under-over wraps. When I add arrows to mark how I enter each wrap, I find that the arrows for 10 wraps enter from opposite sides and the arrows for two (the central ones in the figure) enter from the same side (This result is the reverse of what I found for the "ten men.")

Do the arithmetic relations hold up for other figures? I leave it to you to apply them to the "lightning," a zigzag figure originated by the Navahos. Begin (it is almost needless to say) with the Navaho opening. Bend each thumb away from you so that it passes over two string sections. With the back of the thumb pick up the next section from below. Pull the string by returning each thumb to its original position.

Bend each middle finger toward you, passing it over one string section. With the back of each middle finger pick up the next section from below. Pull the string by returning the middle fingers to their original orientation. Bend each ring finger toward you, passing it over one section. With the back of the finger pick up the next section from below. Pull it by returning the ring fingers to their original orientation.

Now bend each little finger toward you, passing it over one string section, and with the back of the finger pick up the next section from below. Pull on the string by returning the little fingers to their original positions. Bend each thumb away from you and touch its tip on the front string section of the little finger. This move releases the sections wrapped around the thumbs. With a flick of your wrists toss these sections over the other ones. Press your thumb tips down hard on the string sections they touch. To display the figure turn your palms away from you as you spread your fingers and thumbs.

If you find more ways to weave wraps into a figure, I would enjoy hearing from you. Are my rules about the number and nature of the wraps correct? Can you devise general proofs of the rules or find more rules?

Bibliography

FUN WITH STRING FIGURES. W. W. Rouse Ball. Dover Publications, Inc., 1971.

ORDERLY TANGLES: CLOVERLEAFS, GORDIAN KNOTS, AND REGULAR POLYLINKS. Alan Holden. Columbia University Press, 1983.

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