You are well known as the best jeweller in Puzzovania; your shop is always well stocked and your pockets are always bulging.
One day, the local 'godfather' of Puzzovania's organised crime comes into your shop. Naturally, you owe him a favour - who doesn't in these parts? He hands you a diamond the size of a hen's egg and tells you what he requires from you:
"This is for my daughter. She's getting married - you understand? I want her to have the best-cut diamond in the whole of Puzzovania. It's family honour."
You nod mutely, inspecting the diamond in awe. It's the biggest one you've ever seen. You decide it would be more prudent not to ask where the diamond came from.
"So listen, my friend. The diamond needs to be cut to the shape of a convex polyhedron. Any size and shape; that doesn't matter. What matters is that every face of the polyhedron needs to have a different number of sides. My daughter's such a changeable character - never quite the same person twice. I want her jewel to reflect that. Do you think you can manage that for me?"
"Of ... of course, Godfather," you stammer. Once that imposing gentleman has taken his leave, you sit down, still holding the diamond, and wonder how you're going to do it. You need to call in a mathematician ... or possibly a puzzler!
Can you satisfy the Godfather's wishes? If so, construct a polyhedron as required (and prepare your best suit, since he'll probably invite you to the wedding as a reward). If not, prove the non-existence of such a polyhedron (and book your flight out of the country, to somewhere far enough away to be out of the reach of his power).
(Note: this puzzle sounds much better if you read all the Godfather's words in Marlon Brando's voice.)