We analyze a simple random process in which a token is moved in the
interval $A = \{0, n\}$: Fix a probability distribution $\mu$ over $\{1,\dots,n\}$. 
Initially, the token is placed in a random position in $A$. In round
$t$, a random value $d$ is chosen according to $\mu$. If the token is in
position $a\ge d$, then it is moved to position $a-d$. Otherwise it
stays put. Let T be the number of rounds until the token reaches
position $0$. We show tight bounds for the expectation of $T$ for the
optimal distribution $\mu$, i.e., we show that $min_\mu\{E\mu(T)\} =
\Theta((log n)^2)$.  For the proof, a novel potential function argument
is introduced. The research is motivated by the problem of
approximating the minimum of a continuous function over [0, 1] with a
'blind' optimization strategy.