Stretch

Two players create a bipartite graph between two sets of nodes of size N:

• player 1 clicks any node on the left
• player 2 answers by clicking an unused node on the right.
• player 1 may reuse left-nodes, but Player 2 may never reuse a right-node.
• a parameter k sets the difficulty of player 1 winning the game.

A set R of k many right-nodes is a k-stretch if r< m where

• r is the distance between the smallest and largest node in R
• L is the set of left-nodes that have edges with nodes in R
• m is the minimum distance amongst any pair of distinct nodes in L.

Player 1 wins as soon as the there is a k-stretch in the graph within N/2 completed stages, In this case:

• the nodes exhibiting a k-stretch are highlighted by a red border (call them "red nodes")
• yellow edges connect two consecutive red nodes on the left with the smallest and largest red-node on the right

so that the yellow edges from left to right approach each other (this witnesses a k-stretch).