How can we see if this technique works? There are two methods try to know something: deduction and induction.
First a little deduction. Let's try to be explicit about the theory behind this technique.
It's safe to assume that the job will NOT consist mainly of cranking out binary tree inversions on whiteboards while being watched over. So obviously we're hoping to make a correlation with something else. Assuming the candidate was not tipped off and learned this particular puzzle, perhaps we are correlating to an ability to rapidly create novel solutions of long-solved algorithms without reference tools.
But is that what the new hire will be doing? Probably not.
We could continue down this path, identifying ever more removed correlations until we get to something that the job actually demands. This probably involves solving hard problems like naming things. [1] But by now our theory stands on pretty thin ice indeed.
In any case, all of this deduction is theory making. It's not knowledge until we attempt to falsify [2] it via induction. The human mind constantly induces hoping to verify our deductions. We reason, observe, conclude and repeat. We're good enough at it to survive, but that's about it. Lucky for us, science came along. Today's technical hiring is at best alchemy.
A interesting company called TripleByte [3] is trying to apply induction (first for YC companies). They specially shun on white board coding and puzzle solving tests in general. I will be interested to see how they fare and whether their learnings are adopted more broadly.
[1] http://martinfowler.com/bliki/TwoHardThings.html
[2] http://en.wikipedia.org/wiki/Falsification
[3] http://techcrunch.com/2015/05/07/triplebyte/