I was asked to solve this riddle with details for a Mystery Fiction class I am taking a UC Berkeley Extension. I am not including all the details, but these are my possible solutions. If you have new solutions, do share.
A man is found in Rajasthan desert, dead of a broken neck. He is naked. There are no footprints in the sand around him. Besides the dead body, the only other item that does not belong to the desert environment is one-half of a matchstick, unlit.
If the man voluntary jumped: The aliens had abducted the man. As aliens don’t wear clothes, the man was naked. The naked man jumped from the alien ship on the desert. The aliens though the sandy desert will make a soft landing. When the man jumped, he broke his neck. A bird flew by and dropped one-half on an unlit matchstick near the man’s body.
If the man accidentally fell: The man and his lover were in a hot air balloon flying over Rajasthan desert. They have sex so they are naked. The man stands to stretch. He fails to keep his balance and falls down. Due to the force of his fall, he breaks his neck and sand flies in all direction. An old one-half unlit matchstick that was buried deep in the sand dune loosens and surfaces next to the man’s body.
If the man was pushed: The man and his wife were in their private plane flying over Rajasthan desert. The man removes his clothes to get ready for shower. He is an ex-smoker. He puts the end of an unlit matchstick between his front teeth. His greedy wife enters with a gun in her hand. In his fright, the man involuntary bites the matchstick breaking it in two parts. He swallows the lower part of the matchstick. He pretends to cough, covers his mouth with his left palm, and hides the one-half of the unlit matchstick parallel between his index and middle finger. His wife waves the gun at him. She asks him to raise his hands and keep them on top of his head. His wife puts the gun on his temple and propels him towards the exit. The private plane’s pilot is her lover. The pilot opens the exit door. The wife pushes the man. Naked, the man starts falling down. In the course of his fall, one-half of the unlit matchstick flies out from between his fingers. Matchstick and the man fall on the desert. The man breaks his neck on impact.