

A paradoxical math ratio - strategy
http://mindyourdecisions.com/blog/2011/02/01/broken-sticks-puzzle-and-a-seemingly-paradoxical-ratio/

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pmjordan
_"This itself is a rather surprising result: Euler’s constant e comes out of
nowhere!"_

Is it? There's an integral with a 1/x term[1]. How exactly is it surprising
that logarithms start showing up? I realise integrating by hand is a pain, but
surely it's a good idea to at least _think_ about the integral expression
before typing it into Wolfram Alpha. (EDIT: to clarify - the author takes
integration for granted, yet is surprised when the integral evaluates to a
logarithm; this makes little sense to me, integrating polynomials is the first
thing they teach you)

[1] 1/(1-x) is transformed into 1/y in a pretty straightforward substitution.
When you do that with our actual integrals, you find yourself with 2 identical
integrals, as you well should, given the symmetry. My solution was actually to
look at the distribution of breaks already sorted into short & long, which
means doing only one of the integrals (short sticks are 0..1/2) and divide by
the range, since we're taking an average (i.e. multiply by 2).

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ez77
To get really blown away, consider (again if necessary) that the average
spacing between neighboring primes (primes!) less than N _roughly approaches_
ln N [1]... Stunning.

[1] <http://en.wikipedia.org/wiki/Prime_number_theorem>

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archgoon
I think the bonus question is more counter-intuitive, not only is the inverse
ratio not the reciprocal, it doesn't exist.

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michael_dorfman
Exactly. There's nothing paradoxical about this at all, just a bit counter-
intuitive.

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archgoon
Well, he does say 'seemingly paradoxical', and look what happens when you try
and express this (naively) in English:

"The shorter stick is typically (about) 2/5 the length of the longer stick,
but the longer stick is typically much more than 3 times the length of the
shorter stick."

To be honest, I can't figure out how to summarize the two results in English
correctly (short of saying 'the mean of x/y~=2/5, but the mean of y/x does not
exist, where x is the shorter and y the longer length')

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_delirium
There've been some psychological studies on averageness, and I believe one
common case (though I'd have to look it up) is that people think of "typical"
as a representative example in the middle of a distribution, something like a
peak of a distribution and/or a median, and then imagine the distribution as a
cloud of "stuff on either side of that".

So the intuitive answer to the question would require there to exist a
"typical" stick. Once you pick one specific stick, of course things work
intuitively: the stick's dimensions are _x=A, y=B_ , its shorter/longer ratio
is _min(A,B)/max(A,B)_ , and its longer/shorter ratio is _max(A,B)/min(A,B)_.

An interesting question might be whether there's a definition of "typical
stick" that in any useful way provides the answer that's intuitively wanted
here.

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wlievens
Our brains aren't exactly wired to handle uniform distributions too, I guess.
Nature is full of normal distributions anyway.

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wlievens
It's a funny trick, but it touches on something that I can't help but feel is
completely nonsensical: repeatedly picking random numbers from an infinite
set.

It's basically a longer version of the following trick:

1) Pick a random real number in the range [0, 1]

2) Pick another

3) What is the probability that they are identical?

For another example, see:
[http://en.wikipedia.org/wiki/Bertrand_paradox_%28probability...](http://en.wikipedia.org/wiki/Bertrand_paradox_%28probability%29)

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ars
It seems to me the probability that they are identical is zero (or at least
zero at the limit).

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wlievens
My hunch is that it's meaningless at the limit.

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mhb
I like this alternative problem statement (not for the ratio part): If you
break a stick in two places, what is the probability that the pieces can make
a triangle?

There is a nice geometrical solution which is described here: [http://www.cut-
the-knot.org/Curriculum/Probability/TriProbab...](http://www.cut-the-
knot.org/Curriculum/Probability/TriProbability.shtml#Explanation)

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3pt14159
This is pretty basic math. A fun puzzle for high school students maybe, but
anyone remembering any of their comp sci or engineering courses should be able
to solve this very quickly.

~~~
AliCollins
Though it's a nice bit of interesting Maths for a Tuesday afternoon!

