I don't know why you'd ever want to, but in case the situation ever comes up, bite it at the nerve center between the eyes.

As opposed to inhumanely killing octopii by pulling them inside out.

This method is faster, less fussy and makes the octopus easier to prepare for the table.

It's better if the octopus is dead when you buy it. It may not be as fresh, but it also won't cling to your arm.

An octopus you buy at the store is killed by being rotated in a cement mixer.

I can confirm that kessenich's technique works. In the long-gone days of childhood, my father used to take me octopus fishing in the sea of Puglia.
We would snorkel for hours, looking into crevices, cracks and generally anywhere where the wily octopus may hide.
Having found one, he or I would hyperventilate and spike the creature with a trident.

Of course, at this point the octopus was very far from dead. Which means that the biting technique had to applied - this is more easily said than done, when you are bobbing up and down in a choppy sea, and the octopus is doing its level best to get off the trident and into your mask.
But (especially at sea) there is no other way of killing the beast. And let me remind you, octopuses have a strong beak (much like a parrot's), and they do use it.
After the killing business is done, you should turn the head inside out and rip out all the entrails. A cloud of scavenger fish will appear out of nowhere and feast on tasty morsels. The wheel of life and all that, once again apparent.
There are moral lessons in everything, even in killing a mollusk.

As for the why would you ever want to do it: to eat it, of course. Preferrably cooked in a soup, with potatoes. And don't forget to tenderize it before.

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