This three-ball juggling is common, but Juggling can¬†use a lot of complex patterns. How should we organize these patterns so we could talk about them? It was not until the 1980s that mathematicians and jugglers started to solve this problem. The result was SITESWAP. Hope your students love it ūüėČ


Siteswap is the notation to record juggling moves. Here we see the pattern progressing from left. Hands are alternating (red is right Рblack is left.)

First a red ball is tossed high in the air. Then a yellow ball tossed back and forth between the two hands before also being shot high in the sky. The number associated with each throw tells¬†how high it goes. A “1” is a fast toss¬†from one hand¬†to the other.


The right and left hands don’t really need to be identified. We just understand that all siteswap routines use alternating hands.

Here is how we test if a pattern of non-negative integers is a siteswap pattern. First write down the integers repeating them again and again.

Choose a ball and see how it will be tossed. We ignore all the other balls.

The first and second tosses were both “6.” To find the next toss, just look at the number where the ball has now landed.

Another 6…

And another 6…

That was the blue ball. Now let’s choose a colour for the second ball.

Choose the first free hand and make the indicated jump. Here the jump indicated was “1.”

The yellow ball continues by jumping 2,1 and 2.

Now the yellow ball is finished.



The last ball fills all remaining hands. We have a siteswap pattern!



Here is a similar siteswap pattern… (621)¬†above…¬†(612) below.



How does this siteswap pattern differ?

It starts out the same…




The yellow ball starts with a two toss…



Then gets tossed 6…


…and continues being tossed as a 6 forever thereafter…


The next ball chooses the first free hand to start. This¬†ball gets a one toss… but then something ugly happens: The juggler is being asked to catch two balls in the same hand at the same time. At the fourth hand grab we fail. This is not a siteswap pattern after all!

Students should make conjectures about some things that might¬†help determine if a repeating¬†number¬†pattern is a siteswap pattern. Often students find this intimidating. Sometimes I ask students to tell my a conjecture that’s not true. That does not work. This loosens them up.

Conjectures from my students (grades 2-9):

  • Any¬†positive integer¬†repeating is a siteswap.
  • Any (odd,odd) pair like (3,5) is a siteswap pattern.
  • Any (odd,even) pair like (3,4)¬†fails.
  • Any pattern – no matter how long – that has¬†n and n-1 consecutively will fail… so (?,?,?,8,7,?) will always fail.
  • If the average is an integer then the pattern is a siteswap pattern.
  • Any¬†staircase pattern like (1,2,3,4,5,6,7) is a siteswap pattern.

The last two of these conjectures were false. The students should try to find examples that show they are false.

Here is an organized list of siteswap patterns: 1, 2, 3, 4, 31, 5, 6, 42, 51, 312, 411, 7, 8, 53, 62, 71, 3122, 4112, 5111, 9, 423, 441, 522, 531, 612, 711

Your class should discover by themselves that 531 is the same as 153 is the same as 315 is DIFFERENT from 135 (which is not a siteswap pattern). Did the class want to include 531, 153 and 315 all on the list? Maybe, but they should try to be lazy and realize that that is redundant ūüėČ

This is a sample puzzle sheet. Students should use color or be VERY accurate.

Download puzzle sheets here.

Go to a siteswap simulator here.


(Paul Klimek,1981)

Siteswap is a notation to record juggling patterns. It is so cool that it is essential in every elementary students exploration of patterns.

Older students should also be encouraged to jump in to explore the algebra…

Download puzzle sheets here.

Go to a siteswap simulator here.


Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app. 

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist ūüėČ

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos ūüėČ For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)