# Math puzzle: Filling a bath tub with increasingly smaller containers

Originally published at: http://boingboing.net/2017/05/25/math-puzzle-filling-a-bath-tu.html

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This would be interesting, if only it included the puzzle.

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It’s Zeno’s Paradox applied to bathtubs.

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It’s a meta-puzzle: what was the intended puzzle that Mark left out? I’m going to guess maybe the “certain pattern” of containers getting smaller and smaller is that their sizes follow the harmonic series: 1/2, 1/3, 1/4, 1/5… Since the series diverges, amazingly you can fill any size of bath with enough containers, unlike, say, the geometric series 1/2, 1/4, 1/8, 1/16… which will fill a bath of volume 1 when you have added a countable infinity of containers of water, and no larger bath. Anyway, to compute a partial sum of the harmonic series you can use calculus as suggested by the problem. One answer is here: https://math.stackexchange.com/questions/451558/how-to-find-the-sum-of-this-series-1-frac12-frac13-frac14-do .

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Although the question posed is a bit vague (how long does it take to fill the tub if you fill it using a series of containers that decrease in volume in a regular way? Well, that depends on the pour rate, doesn’t it?) he then goes on to talk about upper and lower bounds on the number of pours. So if you’re aiming to fill, but not overflow the tub, it becomes Zeno’s paradox.

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What about a bathtub you can completely fill with paint but never completely paint?

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To cheem: no, it’s not Zeno’s paradox. The upper and lower bounds of the problem come from the approximation of the sum of discrete quantities by the area under a curve (this is where calculus comes in). If you compare the picture in the top-ranked answer here https://math.stackexchange.com/questions/451558/how-to-find-the-sum-of-this-series-1-frac12-frac13-frac14-do , which is an overestimate, to the picture here, http://mathfactor.uark.edu/images/harmonic2.jpg , which is an underestimate, you can see how the problem’s requested upper and lower bounds come into play.

Zeno’s paradox would be relevant if we were dealing with the sum of a geometric series, 1/2 + 1/4 + 1/8 + 1/16…

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This is the question that he forgot to link to.

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Assuming a big enough bath tub and (progressively) small enough containers, evaporation may take over and you’d never fill up the tub.

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It shares some aspects with Xeno’s paradox, insofar as they are both about limits, but these are different beasts. Some counter-intuitive results I can think of with respect to boundaries include th following. Clearly, the tub can be filled using a single container if that container is the size of the tub, and as the size of the container gets larger, the proportion of the container used for the tub goes to 0, so in the limit the number of pours approaches 0 as the container size increases. On the other hand, the proper geometric series of container sizes will fill the tub exactly using an infinite number of containers, which sort of means it will never fill, but you can get arbitrarily close to full if you have enough time. For a convergent progression that starts with a container that is too small, it will never fill, regardless of how many containers you use–even if you fill it forever. And then there is the result @avunculoid mentions, that a divergent progression will fill any tub in finite time, no matter how large.

Other progressions are well known that consecutively add and subtract values in an ‘alternating series’., +1, -1/2, +1/3, etc. With the right proper series, you can use the tub to calculate many functions, including logarithms, trigonometric functions, etc. And if the containers are arranged according to a Taylor series, you can approximate to any degree of precision any real-valued function.

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When I first read the headline it looked like someone was wanting to fill a bathtub with increasingly smaller containers instead of filling it with water by using increasingly smaller containers. I had an image of a Russian nesting bathtub.

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And on the topic of Xeno’s Paradox…

An infinite number of mathematicians walk into a pub. The first mathematician says “I’ll have a pint, please.” The second mathematician says “I’ll have half-a-pint please.” The third mathematician says, “I’ll have a quarter of a pint, please,” but then the bartender interrupts by saying “You guys are assholes”, and pours two pints.

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Or with the pun, “You guys need to know your limits.”

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If the bathtub is n cups, it takes approximately e^n cups.

See: H_n = nth harmonic number.

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The lower bound is easy, it’s the first example where all cups are the same size, 480 pours. If one wants to quibble they have to reduce in size, let each cup be smaller by (1-e) where e --> 0

The upper bound is pretty obviously unlimited. It’s trivial to construct a series that begins at 1 and approaches 480 asymptotically. What’s hard is bumping that number up to aleph-1.

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Which is how we got the Salton Sea:

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