  Sinai's walk is a recurrent one-dimensional nearest-neighbor random walk in
random environment. It is known for a phenomenon of strong localization,
namely, the walk spends almost all time at or near the bottom of deep valleys
of the potential. Our main result shows a weakness of this localization
phenomenon: with probability one, the zones where the walk stays for the most
time can be far away from the sites where the walk spends the most time. In
particular, this gives a negative answer to a problem of Erd\H{o}s and
R\'{e}v\'{e}sz [Mathematical Structures--Computational
Mathematics--Mathematical Modelling 2 (1984) 152--157], originally formulated
for the usual homogeneous random walk.
