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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,...,n\$.Initially, the token is placed in a random position in $A$. In round $t$, arandom value $d$ is chosen according to $\mu$. If the token is in position$a\geq d$, 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 tightbounds for the expectation of $T$ for the optimal distribution $\mu$. Moreprecisely, we show that $\min_\mu\{E_\mu(T)\=\Theta((\log n)^2)$. For theproof, a novel potential function argument is introduced. The research ismotivated by the problem of approximating the minimum of a continuous functionover $[0,1]$ with a ``blind'' optimization strategy.

Tight Bounds for Blind Search on the Integers