Uploaded March 2018 | Updated September 2026, 2 weeks ago

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SM64 TAS Subreddit: reddit.com/r/SM64TAS
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This is an old problem that Jongyon and I came up with around a year or so ago. Although he was more interested in, I guess, the more practical uses, I was more interested in the mathematics.

This problem isn't easy to solve, but if it is requested, I am happy to go through the mathematics in another video. But it wouldn't be too easy to understand for some viewers.

This problem doesn't have a nice clean formula, by the way. It is a multi summation function (equal to the number of dimensions, by the way) that, more or less, has an IF statement in the middle (or multiple for multi-dimensions), along with the GCD(x,y) (or GCD(x,y,z,w,v,...) for.. ok, you get the point). Neither the "IF"s, nor the GCDs can be expressed with mathematics in a formulative sense, making this problem almost impossible to have a CLEAN answer for.

Anyway, here are the .SVGs and .PNG file exports of (0,0) to (128, 128), and (-128, -128) to (127, 127):
cdn.discordapp.com/attachments/351623933457334282/427809731932061696/Images.zip
^Sorry they're kinda bad. The more the dots, the harder it is to render, so they have BAD overlap

And as promised, here is some of the mathematics that was shared between me and other users in a mathematics forum. I won't explain it, because some of it has to do with a multi dimension version of this problem (e.g. the number of unique angles in 3D, 4D, ..., 99D, etc.). I am only providing it to contextualise the mathematics into something you can see. Note that there may be minor, or large, mistakes.
imgur.com/kSg6o3G
imgur.com/53C46j9
imgur.com/jc3t8L2
imgur.com/eEdIrYC
imgur.com/5PMyuwD

Outro music by Brendan Horn
boyonastringmusic.bandcamp.com

twitter.com/el_man_ton/status/977615744182530053