You are given with a square n x n chess board that is divided into single cells where n is always power of 2.you are also given with an infinite amount of trominos and also each trominos can cover 3 sets.the board has one special cell s on which you are not allow to place any trominos. you are task is to cover the whole board with trominos in such a way that any 2 trominos donot overlap each other and every cell except the cell s is covered by some trominos.the indexing starts from 1 and top left cell is indexed as (1,1)
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We are given a 2n×2n2n×2n chessboard and we must prove that we can cover this chessboard with gamma shapes leaving only one square uncovered.
A gamma shape is:

So to solve this I proceeded with induction.
Here is my proof:
For n=1n=1 we have a 2×22×2 chessboard and if we remove square then a gamma shape remains so we can cover this and leave exactly one uncovered. Now suppose that for some n∈Nn∈N we can cover the 2n×2n2n×2n chessboard with gamma shapes living exactly one square out so we can cover it leaving a corner out.
A gamma shape is:

So to solve this I proceeded with induction.
Here is my proof:
For n=1n=1 we have a 2×22×2 chessboard and if we remove square then a gamma shape remains so we can cover this and leave exactly one uncovered. Now suppose that for some n∈Nn∈N we can cover the 2n×2n2n×2n chessboard with gamma shapes living exactly one square out so we can cover it leaving a corner out.
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