Batch 2 - Class 19 - Paradoxes

• Answer: Odd sided n-gon - second guy wins; Even sided n-gon - first guy wins. Invariant is that if there are odd number of even sided n-gons at any stage, then first guy wins, else second

• Russel paradox

  Barber's paradox - In a town, a barber shaves everyone (and only those) who don't shave themselves. Who shaves the barber? https://www.youtube.com/watch?v=GpVRePLMLbU http://en.wikipedia.org/wiki/Russell%27s_paradox (Joke Answer: the barber is a woman!)Librarian's paradox of making a catalog of all books which don't refer to themselves

  Infinite rooms hotel - One day the hotel is full with infinite guests (1) One man walks in (2) Infinite number of men walk in

  (1) Move up everyone one room; (2) Move up person in room x to room 2xhttps://www.youtube.com/watch?v=Uj3_KqkI9Zo

  (MartinShCol - 7.1) A square has a fourth of it removed from a corner (L shape remains). Cut it to divide it into 4 congruent figures. Now take a regular square - can you divide it into 5 congruent figures?

  Bertrand paradox - Probability - Equilateral triangle in a circle. Probability of random chord has length longer than side of triangle (1) Chord rotation around one of the vertices => 1/3 (2) Make two triangles as star => 1/2 (3) Smaller circle inscribed in triangle - if mid point is outside then chord is short, else large - proportionate areas imply 1/4. 

  Resolution is in precise definition of "random". Homework: Do the same thing with square inside a circle - come up with three different answershttps://www.youtube.com/watch?v=uI2FnUmBeeo

  (Joke) Why is 6 afraid of 7

  Because 7 8 9
• (Joke) 5q + 5q =?

  Thank you; "You are welcome"
• (MartinColl - Pg 580) Newcomb's paradox - Two boxes one always has $1000, the other may have 0 or $1M. You are allowed to choose the second box, or both first and second. Which one? (Answer: both). Now there is an oracle who looks into your eyes, predicts which one you are going to choose - if you are greedy, the oracle puts 0 in second box, if you are not they reward you by putting $1M there. Orcale is "almost always correct". Give both arguments...  

  https://www.youtube.com/watch?v=aR5GYeZkgvYOne resolution is that due to inherent contradiction, such a predictor cannot exist.

  
  
• Grandfather's Paradox/ Mad Scientist paradox - A scientist travels back in time machine and kills himself before the time travel happened. Is this possible?

  https://www.youtube.com/watch?v=UJNhLDwj6kI