Decoding the Complexity of Amazons

Ever played a game that's both thrilling and incredibly complex? Welcome to Amazons, a board game that merges elements of Chess and Go. Scientists have been using it to study game theory and artificial intelligence. A while back, a researcher named Buro discovered that even simple endgames in Amazons are as challenging to solve as problems in a category known as NP. But what about the entire game? Let's dive in. We found that figuring out the outcome of an n x n Amazons game is extremely difficult. In fact, it's what we call PSPACE-hard, which means it's as tough as some of the hardest problems out there. We did this by connecting it to another incredibly challenging problem in the world of logic, called two-player formula games.

You might think, "But what about Chess and Go? Aren't they difficult too? " Absolutely, but Amazons has a unique feature. The number of moves in a game is limited, unlike Chess and Go. This makes Amazons part of a special group of games, like Hex and Othello, that are both challenging and have a set number of moves. Our work also provides another way to show that simple Amazons endgames are just as hard as those NP problems. However, there's a catch: we still don't know exactly how tough Amazons is when only a few amazons are on the board. That's a riddle for another time!

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