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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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  • $\begingroup$ See matheducators.stackexchange.com/questions/220/… and math.stackexchange.com/questions/423513/… $\endgroup$
    – quid
    Commented Oct 20, 2014 at 10:45
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    $\begingroup$ This question might interest you. $\endgroup$
    – Git Gud
    Commented Oct 20, 2014 at 10:46
  • $\begingroup$ I always liked the proof of $\frac{d}{dx} x^n = nx^{n-1}$ using induction. It's easy and prettier (to my taste) than using Newton's Binomium. $\endgroup$
    – Krijn
    Commented Oct 20, 2014 at 10:50
  • $\begingroup$ I like the proof of the Fibonacci matrix identity. Very simple but nice. $\endgroup$
    – John Smith
    Commented Oct 20, 2014 at 11:07

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