Suppose for a moment that there was no more food to be had anywhere; that the planet was in total famine. How long would humanity survive by cannibalizing each other?
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There are reports; unverified, that sanctions in places like N. Korea have forced some people to resort to cannibalism. This may seem like a question that is not normal to ask, but there are instances in history where this kind of thing happened.
Well, if people are organised in a way that the population would half everytime they eat eachother, and let's say; one person can survive up to a month per human consumed, the last survivor would be standing alone after 3 years.
It would be great if we could work out answers to total famine, and there are ways for sure, but I doubt that the resolve exists to implement them.
So then, in a nutshell, just short of a lunar month. This makes a lot of sense and confirms my suspicions concerning a huge natural disaster like a volcanic winter or major asteroid impact that shut of photosynthesis resulting in mass starvation.