do the math!: juggling with numbers

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Page 1: DO THE MATH!: Juggling with Numbers

DO THE MATH!: Juggling with NumbersAuthor(s): Erik R. TouSource: Math Horizons, Vol. 21, No. 3 (February 2014), pp. 5-7Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/10.4169/mathhorizons.21.3.5 .

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Page 2: DO THE MATH!: Juggling with Numbers

Erik R. Tou

If you love puzzles, as I do, you might already know the “word morphs” game. The rules are simple: You are given a starting word and an ending word, and must gradually change (or, morph) the starting word into the ending word in

stages. At each stage, you are allowed to change a single letter from the word, and the result must be a valid word. For example, suppose you want to morph the word “bird” into “park.” Here is one way to do it:

BIRDBARDBARKPARKSimple, right? Well, it turns out that one of the foun-

dational results in the mathematics of juggling relies on a numerical variant of the word morph game. But to understand that fact, we need to explore how to make juggling numerical.

Juggling has a long history, with the oldest known depictions appearing in one of the ancient Egyptian temples at Beni Hasan (c. 1994–1781 BCE). Georg Forster, a Prussian scientist who accompanied Captain James Cook on his second voyage to the Pacific Ocean (1772–1775), writes of Tongan women who could juggle up to five gourds at a time. During most of this long history, juggling was the province of entertainers and artists, though there was some overlap with other intellectual pursuits, including mathematics. Only in the 1980s did jugglers develop a way to keep track of different juggling patterns using a numerical code, now known as a siteswap.

To understand the siteswap, I invite you to try a thought experiment. Imagine a juggler is standing in front of you and is juggling three balls in a uniform way, free of tricks, gimmicks, and flaming torches. Now close your eyes (in your imagination, that is). You will hear a regular sequence of “thuds” (or, beats) as the balls hit the juggler’s hands—left and right in alternation. The pattern you may be visualizing can be seen, beat by beat, in figure 1.

Imagine now that you open your eyes and follow the motion of a single ball. Jugglers define the height of a throw as the number of beats that occur between the time the ball is tossed and the time it lands (including the landing beat itself). Be careful here—this version of

height is not directly related to the physical altitude of the ball. Rather, height has more to do with the tempo of the pattern being juggled. Also, a throw of height 0 cor-responds to a skipped beat; no ball is caught or thrown on that beat.

Since most juggling patterns repeat themselves at some point, it is enough to describe a pattern by listing the heights of the throws up to the point at which they repeat. For example, if a juggler throws each ball to a height of 3, the pattern 3, 3, 3, 3, . . . would be denoted by a siteswap of (3). This pattern is shown in figure 1. The siteswap (b) is called a b-ball cascade pattern and is one of the most common juggling patterns. Two other common three-ball patterns are (531) and (441).

Once we have a list of throws, it is possible to con-struct an arc diagram in which each beat corresponds to a dot and the throws are drawn as arcs. Think of the arc diagram as a musical score: The arcs and the dots tell us what is happening at each moment of time. Figures 2, 3, and 4 show the arc diagrams for the

do the math!

Juggling with Numbers

Stefan Paridaen, dejongleur.be

www.maa.org/mathhorizons : : Math Horizons : : February 2014 5

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Page 3: DO THE MATH!: Juggling with Numbers

6 February 2014 : : Math Horizons : : www.maa.org/mathhorizons

siteswaps (3), (531), and (441), respectively.The length or period of a juggling sequence is the

length of the siteswap (so the three previous examples have lengths 1, 3, and 3, respectively). We need to be careful, though: Diff erent siteswaps can represent the same pattern. For example, (531), (315), and (153) all represent the same repetition of throws.

With the siteswap notation in hand, we can pose the fi rst major question for the mathematics of jug-gling: Given a sequence of nonnegative integers, how do we know if it is a valid juggling siteswap? We should be more precise here about the word “valid.” While advanced jugglers can do some amazing tricks, includ-ing the throwing and catching of multiple balls at the same time, we assume our imaginary juggler can throw or catch only one ball at any given beat. When two or more balls land at the same time, we will call this a collision. So, we defi ne a siteswap as valid if there are no collisions.

Two balls will collide if they are thrown at diff erent times, say, beats i and j, and land at the same time, say, beat k. The height hi of the ball thrown at time i is while the other throw has height

So, in order to guarantee no collisions, we must have whenever Because the pattern

is periodic, we need only check the n heights in the siteswap, but to do so we must use arithmetic modulo n. For example, for the siteswap (413), is congruent to modulo 3; indeed, in the juggling sequence 4, 1, 3, 4, 1, 3,…, the third and fi fth tosses collide on the sixth beat. (Draw an arc diagram to verify this for yourself!) This leads us to a characterization theorem for valid siteswaps:

Theorem. A sequence of nonnegative integers is a valid juggling siteswap if the numbers

mod n are distinct for all With this theorem, it is easy to check if a string of

digits is a valid siteswap. Consider the siteswap (54635). We simply compute i + hi for each throw and reduce modulo 5:

Beat: i 1 2 3 4 5

Height: hi 5 4 6 3 5

6 6 9 7 10

mod 5 1 1 4 2 0

Because the fi rst and second numbers are the same modulo 5, this siteswap is invalid. However, if we change the 4 to a 1 we obtain the valid siteswap (51635).

Try it! Show that (825) and (41357) are valid siteswaps and that (4864) is invalid.

This is all well and good, but one crucial question remains: How many balls are required to juggle a given valid siteswap? The following theorem allows us to de-termine, at a glance, the number of balls required.

Theorem. For any valid juggling siteswap, the average of the throw heights equals the number of balls required to juggle that siteswap.

For example, (531), (71), (441), and (66661) re-quire three, four, three, and fi ve balls, respectively. Additionally, we can use the theorem to quickly identify some invalid siteswaps: (2463) is not valid because its average is 3.75.

How do we know that this theorem is true? We can

Figure 1. Six beats of the three-ball cascade pattern.

Figure 2. An arc diagram for the three-ball cascade pattern (3).

Figure 4. An arc diagram for the siteswap (441).

Figure 3. An arc diagram for the siteswap (531). Notice that one of the balls (in red) is always thrown with height 3, while the other two alternate between throws of height 5 and 1.

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Page 4: DO THE MATH!: Juggling with Numbers

prove it using a numerical version of the word morphs game (let’s call it number morphs).

Here are the rules for number morphs: You are given a siteswap, and you must gradually even it out so that the heights in the fi nal siteswap are all the same. At each step, you may interchange any two adjacent num-bers and then transfer a “unit” of height from the right digit to the left digit (we’ll call this process a height swap). For instance, to apply a height swap to the adja-cent pair 62, switch it to 26 and then transfer a unit of height from right to left, yielding 35. It is important to recall that a siteswap represents a periodic sequence, so the last digit in a siteswap is adjacent to (and to the left of) the fi rst digit.

In fi gure 5 we see that a height swap merely switches the landing times of the two balls in question. So a height swap can’t create or eliminate a collision. In particular, a siteswap is valid if, and only if, it is valid after performing a height swap. Moreover, a height swap doesn’t change the number of balls in play and it does not change the average of the heights. Thus, if we win at number morphs, then the original siteswap is valid and the average of the heights is the number of balls required to juggle it.

Try it! Show that if we perform a height swap twice on any pair of numbers, they return to their original values.

Let’s play the game! The siteswap that we encoun-tered earlier, (51635), can be morphed in the following way:

5163551455244554445344444This shows that (51635) is a valid siteswap requiring

four balls.Try it! Play number morphs with the valid siteswaps (825)

and (41357). Try playing number morphs with the invalid siteswap (4864).

We still have not fi nished the proof; to do so we must show that if a siteswap is valid, then it is pos-sible to win at number morphs. So, suppose we begin with a valid siteswap. If we are at a nonwinning stage, then, because the sequence is periodic, there must be a maximal height that is followed by one of lower height

(for example, 51 in the siteswap (51455)). Moreover, the larger height is at least two larger than the smaller one, for otherwise the two balls will collide (in fact, you know that you will never encounter an adjacent pair because that would mean the siteswap is invalid, thus making the original siteswap invalid). Perform a height swap on this pair. Afterward, either the maximal height of the new siteswap is one smaller or there is one fewer of the maximal value. Repeat this procedure. Because the average of the heights remains unchanged, this strategy must terminate in a constant sequence. This gives us a winning strategy for any valid siteswap, thus completing the proof.

Try it! Now that you have seen some of the mathemati-cal tricks that lie underneath the juggling tricks, it is time to juggle. There are exactly three valid siteswaps that have length two and require three balls: (60), (51), and (42). Try juggling these. What are the 12 valid siteswaps that have length three and require three balls? (Hint: six of them involve at least one throw of height zero.)

FURTHER READING

The most comprehensive resource is Burkard Polster’s book The Mathematics of Juggling (Springer-Verlag, 2003).

The Juggling Information Service (juggling.org) is an amazing resource. One important item is Francisco Alvarez’s book Juggling—Its History and Greatest Performers (juggling.org/books/alvarez).

You can have a computer generate juggling anima-tions using siteswaps! A good program is available at jugglinglab.sourceforge.net.

To read jugglers’ opinions of the mathematics of juggling see Gregory S. Warrington’s article “Juggling Performers + Math = ?” in the February 2008 issue of Math Horizons.

More advanced readers can check out Joe Buhler and Ron Graham’s chapter, “Juggling Drops and Descents,” in Mathematical Adventures for Students and Amateurs (MAA, 2004). ■

Erik Tou teaches computer science at Pacifi c Lutheran University in Tacoma, Washington. His research inter-ests include number theory and the history of mathemat-ics. He is a co-director of the Euler Archive, and an author of a recent paper, “A Zeta Function for Juggling Patterns,” which appeared in the Journal of Number Theory.Email: [email protected]

http://dx.doi.org/10.4169/mathhorizons.21.3.5

Figure 5. Replacing 62 (gray) with 35 (green).

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