The game played with red eggs at Greek Easter is called tsougrisma and it involves two players and red eggs.
Each player holds a red egg, and one taps the end of her/his egg lightly against the end of the other player's egg. The goal is to crack the opponent's egg. When one end is cracked, the winner uses the same end of her/his egg to try to crack the other end of the opponent's egg.
The player who successfully cracks the eggs of the other players is declared the winner and, it is said, will have good luck during the year.
There are no rules about which end of the egg to tap first, how to hold it, or how to tap - but I've never seen a "system" that works all the time!
The word tsougrisma means "clinking together" or "clashing." In Greek: τσούγκρισμα, pronounced TSOO-grees-mah.