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The quest for longer hangover February 21, 2009

Posted by keithkchan in fun stuffs.
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Today, I read a very interesting article in Science, the joys of longer hangover (subscription required). What the hell is that? You may heard of this quiz before. Suppose you have N, say 100 identical bricks, you are asked to stack them on each other near the edge of a table so that the farthest end of the brick is as far away from the table as possible.

The answer normal people like me come up with is as follows. Let’s start with the last one. You want to get it as far as possible. So it extends 1/2 of the brick. The second last one can maximally extend 1/4, and so on so forth. You get the half of the harmonic series 1/2( 1 + 1/2 + 1/4 + … + 1/N). This series diverges logarithmically. The stacking is illustrated in following figure.

What’s more surprisingly is that some mathematicians have come up with a better strategy. Look at the following orange stacking. It outplays the conventional answer substantially. More remarkably, the mathematicians managed to transform the problem to some random walk problem and derived a maximum hangover bound 6N^{1/3}. But this is an upper bound and the best strategy they get 0.57 N^{1/3}. It is not sure how close to the upper bound it can be achieved. But this is a big big achievement, from logarithmically divergent to cubic root.

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