However: If you rotate the semi-circle about its diameter, it forms a sphere. Pappus’ Theorem then applies:

(area) (distance COM travels) = (volume of the solid swept out)

(pi/2)(2pi*x) = (4pi/3)

From which x = 4/(3pi) ]]>

For me the most miraculous items in this category seem to be with expected value, where first of all linearity lets you dispense with all kinds of information about covariances, and second of all you can sum a whole bunch of infinite series that come up in expected value calculations by solving a simple linear equation instead.

]]>Rejection sampling: draw a sample from a distribution defined by a probability density function, without finding its inverse cumulative distribution.

Homomorphic encryption: compute with encrypted data without first decrypting it: http://www.americanscientist.org/issues/pub/2012/5/alice-and-bob-in-cipherspace/1

Maximum likelihood estimation of exponential family models by stochastic approximation: no need to approximate the likelihood being optimized, just go straight for its derivatives (easier to estimate).

A nice class of things…I look forward to seeing more!

]]>(denominator-numerator)/denominator

…is irreducible, then so is the original fraction, and if it is reducible, so is the original fraction. Occasionally, it’s easier to eyeball whether the expression above is reducible or not than dealing with what you’re initially given.

]]>I suspect this sort of thing happens a lot in computing integrals (and other cook-bookery type math) where no one algorithm works.

]]>E.g. S = 1 + 2 + … + 2713

2×S = (1 + 2713) + (2 + 2712) + … + (2713+1) = 2713×(2713+1)

So S = 2713×(2713+1)/2 = 3681541

So there is no need to waste one’s time with 2712 additions.