[math-fun] checkerboard version of an e-related dating problem

Cornell mathematician Steven Strogatz thinks this is an unsolved problem but am wondering if there's something being overlooked. Here's the basic -- and solved -- problem. There's a long line of people standing near each other. They're told to hold hands with someone - either the person to their right or to their left, if available, but not both. The ratio of the resulting singles to all the people in line approaches 1/e^2 as the line increases. Now the unsolved version: what if the people are instead in a grid and have up to four people to choose from (but can hold hands with at most 1)?