Collision symmetry

The key insight is that when two ants meet and reverse direction, the system is equivalent to the ants passing through each other without interaction, because the ants are indistinguishable. Under this "pass-through" model, each ant simply continues in its original direction forever. Therefore, the number of ants that reach Priya's end equals the number of ants that were initially moving leftward (toward Priya), and the number that reach Theo's end equals the number initially moving rightward (toward Theo).

Initially, all 40 ants at Priya's end move rightward (toward Theo), and all 80 ants at Theo's end move leftward (toward Priya). Under the pass-through equivalence, the ants that reach Priya are those that started at Theo's end and moved leftward: 80 ants. The ants that reach Theo are those that started at Priya's end and moved rightward: 40 ants. Thus $x = 80$, $y = 40$, and

More formally, let $x$ be the number reaching Priya and $y$ the number reaching Theo. Under the pass-through model, each ant's trajectory is a straight line from its start to the opposite end. Since all ants move at the same speed, the number reaching each end is exactly the number that started at the opposite end. Hence $x = 80$, $y = 40$, and the difference is $40$.