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May 25th, 2017, 11:21 AM   #1
Joined: May 2017
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Intro to Math Proofs - Induction

Hello all,
I am following the MIT OpenCourseware Mathematics for computer science. The current topic is proof by induction. In Lecture 2 at the ~1:00:00 mark, this problem is given: A 2^n x 2^n square can be covered by an L (nxn)shaped tile such that there is one open tile in the center.
n n

Thm: for all n there exists a way to cover a 2^n x 2^n area with a center square (open/missing) using L shaped tiles.
Pf: by induction P(n)= a 2^n x 2^n can be covered by L shaped tiles with a center tile missing.
**** Problem area***
Base Case: P(0)=creates area of size 1. the professor(Tom Leighton) says this is true. that the center is open.
My thinking is the base case fails, because there is no L shaped tile that can cover the area with a missing nxn area, as stated in the theorem.<--- Most troubling
Same goes for n=1 because then the area is 2x2 and while an L shaped tile can cover 3/4 of the area, the one remaining square is on an edge and not the center. But I could see where functionally the thm is met because the tile is used and there is a 1x1 area free/missing.

Is this a case of semantics or do I have a sound argument?


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