There are $N$ buckets.

Each second we add one new ball to a random bucket - so at $t=k$, there are a total of $k$ balls collectively in the buckets.

At $t=1$, we expect that at least one bucket contains one ball.

At $t=\sqrt{2N\ln{2}}$, due to birthday paradox, we expect that at least one bucket contains two balls.

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At $t=f(m)$, we expect at least one bucket to contain $m$ balls.

What is the function $f(m)$?

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Actually, the probability $p$ that there bucket with more than one ball occurs is first matched or exceeded at $t=\sqrt {2N\ln \frac 1{1-p}}$. – Roman Chokler Aug 25 '12 at 4:24
The birthday paradox is closer to $\sqrt{N\sqrt 2}$. The classic one is $23\approx \sqrt {365 \sqrt 2}$ A naive approach says you have $\frac {N(N-1)}2$ pairs, so when this is of the order of $1$ you should expect a match. – Ross Millikan Aug 25 '12 at 4:30
This is not $\sqrt{2N\ln(2)}$ to obtain a 50% chance ? – Xoff Aug 29 '12 at 21:38