We analyze a simple random process in which a token is moved in the

interval $A={0,dots,n$: 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 $ageq d$, 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 $mu$. More precisely, we show that

$min_mu{E_mu(T)=Thetaleft((log n)^2

ight)$. 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.