## Problem of the Day #280: Gift Labeling
*December 24, 2011*

*Posted by Alex in : potd , trackback*

Santa Saketh has hired monkeys to label the many gifts he has. Each second, Saketh’s monkeys randomly write a number in the range $[1, 100]$ and Saketh labels that gift if it has not already been labeled. After how many seconds will the expected number of labeled gifts exceed $50$?

Bonus: the probability a number is written is inversely proportional to its number of digits. For example, $1$ has twice the probability of being written as $12$. Solve the problem with this new information.

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