Thread by @panlepan

I made this GIF w/ @geogebra to show the differents throws 1, 2, 3, 4, 5 when you juggle, aka #siteswap notation. #thread
cc @ColinTheMathmo

You can then combine these throws for a juggling trick which will be coded by the corresponding numbers. eg this is 441. (just the period)

This is 12345.
Note: the 5 is always trown with the blue ball, and
the 2 is not actually thrown: the ball remains in the hand for 2 beats.

The regular juggling with n balls is just coded by n. eg here with four balls you only throw 4s (444444...). So this is the siteswap: 4.

Here siteswap 53. Since the length of the period is even (yes, 2 is even) and you have 2 hands (I guess), one hand throws 5 and the other 3.

This is 501. The 0 means the hand is actually empty at that moment, so you throw twice with the same hand.
Just two balls, but not so easy!

Why not put a third ball when that hand is empty? It's possible, you then have this 531 instead of 501. The blue ball throws the 3s.

I didn't mention the most usual pattern, the regular 3 ball juggling. Just 3. Aka the 8 figure, the three ball cascade...

When the number of balls is odd, the regular juggling pattern is always a cascade where every throw crosses. Here 7.

I hope you got it now.
Can you find out what the #siteswap for this pattern is?
@solvemymaths

The average theorem:
The average of the numbers in a siteswap is the number of balls you need to juggle that trick.
e.g. 53 needs 4 balls

So a sequence whose average is not an integer cannot be a valid siteswap.
eg 523 can't be valid because (5+2+3)/3=10/3

This condition of an integer average is necessary, but not sufficient.
You can juggle 123 with (1+2+3)/3=2 balls, but not 321 as shown here

Later I'll show how you can find all the possible siteswaps for a given maximum height. This GIF I made requires some explanations...