The simplicity and static nature of always having 3 balls within the 2d playing surface is what makes the game so appealing, imo.
(Though I did jump the edge of a ball once and got the NO frown

)
Carom games can use 3 or 4 balls and each player has his one cb for the whole game.
3balls is usually 3 cushion and
4 balls is usually straight rail.
Player must make his ball strike the other 2. In three cushion, one must strike the 2nd ball after contacting 3+ rails with cb to score. In straight rail, one must contact the 2 red balls without contacting the other player's cb with his own cb.
Super simple, infinitely complex.
No ball in hand, no hangers!