Short answers, pulled from the story.
Trigon was an ancient Roman ball game in which three players stood at the corners of a triangle, threw a small hard ball with their right hand, and caught with their left. Each player held between three and six balls at a time, and the goal was to keep them circling continuously at high speed. Assistants called pilecripi kept score and retrieved stray balls.
The name trigon derives from the Greek word trigōnos, meaning three-cornered or triangular. It likely reflects the triangular formation the three players took when standing to play the game. The Roman game may have been adapted from a Greek game also called trigōn.
The pilecripi were assistants who stood alongside the three main trigon players, called trigonali. Their role was to keep score and to retrieve any balls that escaped the playing area. In the standard form of the game, they counted successful passes; in the Satyricon scene, one pilecripus counted dropped balls instead.
The Satyricon by Petronius contains a description of a ball game widely assumed to be trigon, though the text never names it. The character Trimalchio, a bald old man, plays with young slave boys while wearing sandals, never retrieving dropped balls, and having a servant supply fresh ones from a sack. Petronius notes that the scorekeeper counted dropped balls rather than successful catches, possibly as a joke about Trimalchio's poor skill.
Two versions of scoring are recorded. In one, a player lost a point for each failed catch, and whoever had the fewest points at the end lost. In the other, a single dropped ball ended the game immediately, and the player who missed it was the loser. The Satyricon also describes a variant where the scorekeeper counted drops rather than catches.
Trigon required all catching to be done with the left hand while throwing with the right, making reliable left-handed catches genuinely difficult. Players who could catch consistently with their left hand were regarded as admirable because of the skill that required. The game's design, circling hard balls at high speed between three players, made that left-hand reliability the central test of ability.