Problem
COM-B2-M04-P002 Arrows and a Cycle
#2
★★☆☆☆ Level 2 of 5
In a finite set of points, from each point an arrow is drawn to one of the other points. Prove that by following arrows one eventually enters a cycle.
Start from any point and record the sequence of visited points.
Start from an arbitrary point \(v_1\) and always follow the arrow: \(v_1,v_2,v_3,\ldots\). Since there are finitely many points, some point repeats: \(v_i=v_j\) with \(i Then the segment \(v_i,v_{i+1},\ldots,v_{j-1},v_j=v_i\) forms a directed cycle. Hence such a cycle exists.
This may be discussed as a form of the extremal principle: infinite continuation in a finite system forces repetition.