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It's worth noted that this does not work in the other direction - just because the average is a whole number, that does not necessarily mean that the sequence is a legal juggling trick.
Standard example: 432 is not legal, but 423 is.

31 can be called 13.

When I learned the notation, I was told that the convention is choosing among all cyclic permutations the largest in lexicographical order. So "31" rather than "13" should be the norm. All the often-mentioned siteswaps seem to follow this convention.

That's not always the best thing to do. For example, 45141 is a ground state three ball SiteSwap, one of my favorites. If you write it as 51414 it's not longer possible to go directly into it from cascade.

The original complaint was that four is a lot harder than five. This is manifestly false.

On the other hand, if your target is to get to five, then I *still* think it's faster to learn a four fountain, then learn simpler tricks with four, then learn some SiteSwaps with four, then move to working directly on five, rather than working just on doing five.

Yes, different people have different reasons for learning, and different people will learn best in different ways, but trying to do four in one of the harder patterns, and then claiming that four is harder than five, is simply perverse.

Research is to see what everybody else has seen, and think what nobody else has thought.