(This page is intended as a standard link from individual koans, to explain intended use.)
A "koan" is a puzzle that the reader is meant to attempt to solve before continuing. This method attempts to reflect the research which shows that you're much more likely to remember a fact or solution if you try to solve the problem yourself before reading the solution. Succeed or fail, the important thing is to have 'tried first.
Trying to solve koans is particularly important on LW (and some other places), which is that since conclusions about rationality often sound obvious in retrospect, it's hard for people to visualize the diff between "what I believed before" and "what I believed afterward". Explicitly having some idea of this difference is important to learning....