He has a writeup of it here: https://eieio.games/blog/writing-down-every-uuid/

He's using a bijection between the number on the left (which are ordered) and the UUID that's designed to produce something random-looking. The simple fact that bijection cannot produce the same output out of two inputs (or it's not a bijection) means that all the UUIDs are unique.

UUID is a 128-bit value, of which 122 bits are available for actual data. UUID was conceived as a way to mark an intersection point between two dimensions, originally for Apollo Computer's RPC system where the two dimensions were time and hardware identifier (originally originally Apollo hardware serial number, later commonly 802.3 address). The additional bits are metadata telling you that a particular 128-bit value is a UUID, not just a coincidental jumble of bits, and which type of UUID.

This site is specifically about “V4” UUIDs where the two dimensions are random and also-random. If you scroll all the way to the bottom you will see that the last table row is numbered the same value as the maximum 122-bit number, so the site is flipping every possible bit-combination within that space combined with the metadata bits that say “Hello I am a V4 UUID”:

  irb> ::Array::new(122, 0b1).reduce { (_1 << 1) | _2 } => 5316911983139663491615228241121378303

They're not randomly generated. They're generated using a Feistel cipher. [1]

He's ensuring the uniqueness by indexing from 0, and applying the cipher to come up with a more "random" looking UUID.

[1] https://eieio.games/blog/writing-down-every-uuid/#toc:feiste...

They are not, 100% strictly speaking, “ensured”. But they are 128bit numbers, so you have realistically no chance of generating a uuid that someone else has already. Age-of-the-universe type chances of duplicating one.

UUIDs aren't technically unique, they're just designed in such a way that the chance of collisions is very small.

A big part of this is that the possibility space is very large, so the chance of collisions is low. Many UUID versions also determine parts of the UUID via MAC addresses and timestamps, to ensure that different servers are highly unlikely to generate the same UUID.