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I am searching for nice proofs by induction, that can be used to teach. I remember this example, that my analysis professor presented to us in first semester and I am searching for more such easily understood examples.

Theorem: A chess-board with side length $2^n$ for $n \in \mathbb{N}$, where one square in the corner is cut off (see picture for $n=2$), can be tessellated by pieces consisting of 3 squares in an L-shape (see picture).

The piece to tessellate the board

Board for $n=2$

Proof: By induction. For $n=1$, the piece and the board have the same shape. For $n>1$, the board having side lengt $2^n$, divide it into 4 such boards with side length $2^{n-1}$, where one of those smaller boards has its cut-off corner in the corner of the bigger board and the other three have their cut-off corner in the center of the bigger board, leaving 3 pieces in the center, which can be filled by one more L-piece.

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