Can someone give me an example of a challenging proof by induction? I'm about to give my first consultation and the topic is proof by induction. The students are new to calculus and I assume they don't know anything besides precalculus. Can anyone give me a proof by induction which is a bit different, challenging, maybe foreshadows other areas of calculus (derivation or whatever) because the prof who teaches them as well already have shown them a lot of easy ones. I'm glad even if you can give an example where it is hard to see that it can be proved by induction.
Here is one that my math professor showed us: show that for $n\ge 1$ a $2^n\times 2^n$ chessboard with one square removed can always be tiled by "L-shaped" pieces. That is, pieces formed by removing a corner from a $2\times2$ square.
Construct the natural numbers $\mathbb{N}$ by using the Peano Axioms. Denote by $\varphi:\mathbb{N}\rightarrow\mathbb{N}$ the map defining the injective map associating each number to the "next one".

Show that the map $+:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ defined by
$$
n+1=\varphi(n),\qquad
n+\varphi(m)=\varphi(n+m)
$$
ís a commutative associative operation.