  In 1960 R\'enyi in his Michigan State University lectures asked for the
number of random queries necessary to recover a hidden bijective labeling of
$n$ distinct objects. In each query one selects a random subset of labels and
asks, which objects have these labels? We consider here an asymmetric version
of the problem in which in every query an object is chosen with probability $p
> 1/2$ and we ignore "inconclusive" queries. We study the number of queries
needed to recover the labeling in its entirety ($H_n$), before at least one
element is recovered ($F_n$), and to recover a randomly chosen element $(D_n)$.
This problem exhibits several remarkable behaviors: $D_n$ converges in
probability but not almost surely, $H_n$ and $F_n$ exhibit phase transitions
with respect to $p$ in the second term. We prove that for $p>1/2$ with high
probability (whp) we need $H_n=\log_{1/p} n +\frac 12 \log_{p/(1-p)}\log n
+o(\log \log n) $ queries to recover the entire bijection. This should be
compared to its symmetric ($p=1/2$) counterpart established by Pittel and
Rubin, who proved that in this case one requires $H_n=\log_{2} n +\sqrt{2
\log_{2} n} +o(\sqrt{\log n}) $ queries. As a bonus, our analysis implies novel
results for random PATRICIA tries, as the problem is probabilistically
equivalent to that of the height, fillup level, and typical depth of a PATRICIA
trie built from $n$ independent binary sequences generated by a biased($p$)
memoryless source.
