Survey on fusible numbers: https://arxiv.org/abs/1202.5614

Fusible numbers are what you get from 0 by operations \(f(x,y)=(x+y+1)/2\) restricted to \(|x-y|<1\). They model a puzzle of measuring time intervals using 1-minute fuses whose ends you can light when another fuse burns out, and are well-ordered with unknown order type. @jeffgerickson calls his (wrong) conjecture "one of my bugs that really should be better known" and points out the lack of progress since 2012: https://mathstodon.xyz/@jeffgerickson/101043945178816599