If I'm reading http://wardsattic.com/joomla/Download/BellmanForestProblem.p... correctly, the ratio you're looking for is ≈ 2.0471.
In Ward's paper, he summarized Moser's Worm Problem differently (p.7):
In 1966, Moser posed a related problem which roughly asks: “What's the best-shaped hammer for smashing one-inch worms?”
Great visualization!