A probability puzzle

I saw a ‘simple’ puzzle on the internet which I thought I’d have a crack at in an evening. Several furious scribblings on the bus and the sofa later, I finally have an answer. I’m so relieved I can’t help but share the joy.

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Fear of heights

I am a short man. I have an even shorter girlfriend. Odds are, any offspring would be short. Odds are, their partners would be similar or smaller in height, especially if they are male. Does this mean I am destined to be the heir to a slowly shrinking troglodyte race, my pristine DNA squirted from troll to troll until it degrades to a corrosive, fetid broth? Let’s put the thesaurus away and have a look.

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On which shaped planet am I the heaviest?

Continuing on from my last post concerning optimisation and Lagrange multipliers, I came across a neat little paper on the arXiv here, which asks and answers the question: what shape should a planet be to maximise the gravitational force at a given position? This is a fun problem, solved using an extension of the techniques from the last post, namely the use of Lagrange multipliers to optimise a function given some constraint.

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