Here’s an interesting thing I learnt over christmas which I think a very small number of people will find very interesting.
To begin, anyone who doesn’t have a basic understanding of siteswap notation, go get one :) I’ll sumarise in a second but the wiki page explains it relatively well. We only need to concern ourselves with vanilla siteswap for the purpose of this although I suspect (but haven’t been able to find or construct a proof) that a similar result is true for multiplex and multi-handed siteswap.
So, to sumarise siteswap. It is a notation for recording certain aspects of juggling patterns. A siteswap (or juggling sequence) in a string of numbers which record the relative heights a juggler would throw consecutive throws to juggle the given pattern. Hands alternate throws and throw at a regular rhythm The number given is how many beats later the ball being thrown will land (and therefore be thrown again.) As I say this is brief and not meant to be exhaustive so if you’re unsure, check back to the link above.
Moving on to the mater at hand.
One result within the study of siteswap which comes up a lot is what the end of the wiki section of vanilla siteswap begins to talk about and is what I will refer to as ‘The Average Theorem.’ The reason it comes up regularly (between jugglers) is because it is useful because it does two things very quickly.
a) For a known valid siteswap it tells you how many objects are required to juggle it.
b) It provides a quick indication as too whether a sequence of numbers in a valid siteswap or not.
The average theorem essentially says two things, the average of the digits in juggling sequence is the number of balls in the sequence. And hence, if the average of a juggling sequence is not an integer (whole number,) then it can not be a juggling sequence (valid siteswap.)
To see the average theorem in action we can look quickly at an example.
Say someone walked up to us and said 'Is 61314 a valid siteswap?’ we could calculate the average of the 5 numbers given to get the answer (6+1+3+1+4)/5 = 5 and be able to answer maybe. That is to say the average theorem hasn’t ruled it out. It turns out that 61314 is a valid siteswap (can be done a number of ways including the fancy 'Permutation Test’ or by checking the orbits.) Once we know that you can then use the average theorem to you that 61314 is a 5 ball juggling pattern. Yay!
So, that’s the bit that any juggler could tell you, now for the interesting bit…
It becomes obvious quite quickly that the converse of the average theorem is not true. That is to say, if the average of a sequence of numbers is an integer, that does not imply that the sequence is a valid siteswap pattern. This can be shown quickly and easily using a counterexample. 432 has average (4+3+2)/3 = 3 however any juggler will quickly tell you that you cannot juggle 432 (from a cold start all three balls collide on beat 4.)
HOWEVER a partial converse is true.
Given any sequence of numbers which have an average which is an integer. Then there exists at least one permutation of the sequence which is a valid siteswap sequence.
To check back to our example from earlier, 432 average at 3 so a juggle-able permutation must exist and happens to be the very common, jugglers favorite, 423.
For fun I guess you could start averaging you’re shopping bill values (taken to the nearest pound) and whenever you get an integer, find the appropriate siteswap. Oh the fun you could have.
Burkard Polster gives a full proof of this surprising little theorem in his book 'The Mathematics Of Juggling, 2002’ which is in tern based on a proof given by Marshel Hall in 1952 to a theorem about Abelian groups that is is a special case of.
(The only receipt I can find at the moment is for £1.40, £1.40 and £7. How boring…couldn’t get that one wrong if I tried…)