Death by infinity puzzles and the Axiom of Choice

In this video the Mathologer sets out to commit the perfect murder using infinitely many assassins and, subsequently, to get them off the hook in court. The story is broken up into three very tricky puzzles. Challenge yourself to figure them out before the Mathologer reveals his own solutions. Featuring Batman, the controversial Axiom of Choice and a guest appearance by the Banach-Tarski paradox.

The pictures that I used for the Banach-Tarski ball splitting action were grabbed off the brilliant VSauce's video on the Banach-Tarski paradox (https://youtu.be/s86-Z-CbaHA) I mainly did this for easy reference since most people here will have seen this video and in this way would be able to connect easily with what I am talking about here.
I also mention that those sets that get pushed around in the Banach-Tarski paradox are constructed using the Axiom of Choice. Vsauce actually does not mention this although this is really a big deal as far as mathematics is concerned (understandable though since his video was already very long). Here is a link to the spot in the Vsauce video where the Axiom of Choice is envoked (although you have to have a really close look to see how :) https://youtu.be/s86-Z-CbaHA?t=14m2s

There is a very nice TEDed video about the finite version of the last of our puzzles: https://youtu.be/N5vJSNXPEwA The solutions to both the infinite and the finite version are closely related.

Oh and today's t-shirt is from here http://shirt.woot.com/offers/infinite-doughnut?ref=cnt_ctlg_dgn_3

Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.

Enjoy!

Burkard

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