Thanks to commenter Alex Mahdavi who wrote:
Building on your Bridg-it game, I think of this problem in terms of routes. Either there will be a successful route across the river, or there won’t. Now imagine a tall ship is coming down the river, whose mast is too tall for the ship to pass under any bridges. From the ship’s perspective, it is also looking for a successful route- a navigable route downstream past the islands. Similar to the two sides in Bridg-it, there can either be a successful land route, or a successful water route, but there cannot be both (any land route would block any water route, and vice versa).
Yes, thank you! This is exactly what I was trying to say (but didn’t really spell it out at all) when I brought up Bridg-It and why I suspected that the answer would be 50%. Bridg-It says right on its box “Never a tie – always a winner” which means once all the bridge pieces are played, there will either be a North-South route or an East-West route, but not both. Because of this I suspected immediately that the answer must be 50%.
Here’s a better picture of Bridg-It:
Once a piece is played, it is basically saying that either a bridge has survived the storm, or it hasn’t. Either Alex’s tall-masted ship can pass through that space, or a person can walk across the bridge, but not both. From both the North-South and East-West perspectives, the game is played on a grid of NxN+1 “islands.”
This game had been donated to my elementary school along with a bunch of other games, some with pieces missing, that lived in a box in the back of the classroom. It was already old even back then. We played it for a while before we all kind of figured out that whoever goes first wins, so it was neither “new” nor “unpredictable.” Yes, it was flawed from a turn-based-game-play point of view, but as a mathematical model, it was balanced, so 50% seemed likely. Despite its flaws, it did (inadvertently?) introduce us to some logic and geometric concepts. If nothing else, it gave me a clue about a logic puzzle many years later. Board games! They’re important!
UPDATE TO THE UPDATE: I think I’ve got the C++ formatting working properly on the River post after many attempts. WordPress’s editor will sometimes randomly replace all special characters like ” and < and > with their HTML-escape code sequences like " and >. It’s awesome! Every time you edit a post, you have the seemingly random chance that all your code will be turned to gibberish. Anyway, I think it’s okay now, so I won’t be touching that post ever again.
Also, for some reason I thought the title Bridg-It had an exclamation point, but apparently that was my own enthusiasm for the game clouding my perceptions. I fixed it in this post, but I’m not editing that last one.