existence of two different atlases in $S^{7} \subseteq \mathbb{R}^8$ that are not diffeomorphic This question was part of exercises in my manifold course which I am taking online.

Question : There exists two different atlases on $S^{7} \subseteq \mathbb{R}^8$ that are not diffeomorphic. ( exotic sphere)

Attempt: I have to show that two atlases exists by constructing them. This seemed to be a really hard job.
On the other hand , I thought if I could assume that there doesn't exists two atlases that are not diffeomorphic ie each atlases is diffeomorphic to other atlas.
So, { $(U_{\alpha},\phi_{\alpha})$} , {$ ( V_{\beta} , \psi_{\beta})$} are diffeomorphic. But, I am really sorry that I am not able to move forward from here.
I have been following Frank Warner's  Introduction to differntial manifolds.
 A: Just to be clear, this is in no sense an "exercise" unless you were specifically fed a bunch of results leading up to it. Exotic $7$-spheres were first constructed by Milnor in his 1956 Annals paper On manifolds homeomorphic to the $7$-sphere, and as far as I know this was the first construction of an exotic smooth structure to appear in the mathematical literature. This was not at all an easy or trivial result. I have no idea what whoever assigned this as an "exercise" had in mind.
A: There's $28$ $S^7$'s and they form an abelian group.   John Milnor discovered them, and probably won a Field's medal for it.
They're homeomorphic but not diffeomorphic.
In any dimension other than $4$ the differentiable structures on $S^n$ form a finite abelian group. (They always form an abelian monoid.)
Here's one of his first constructions.
Now I am paraphrasing (because I find this interesting).  Apparently he proved it wasn't diffeomorphic to the standard $S^7$ by showing it isn't the boundary of any smooth $8$-manifold with vanishing $4$-th Betti number.  Alternatively it has no orientation reversing diffeomorphism.
He used Morse theory to prove they're homeomorphic.
How many, or if,  exotic $4$-spheres exist is an unsolved problem.
By the generalized Poincare conjecture all homotopy $n$-spheres are homeomorphic to $S^n$.  (John Stallings, whom I had the pleasure of studying under as an undergraduate, did dimension $6$.)
