… And Scrabble Proved PSPACE-Complete

Having been invented in the US in the mid-20th century, Scrabble is now available in dozens of languages and sells in numbers measured in hundreds of millions. That makes it one of the most popular games in the world.  

That has naturally piqued the interest of game theorists. “Since Scrabble is such a successful game, it becomes a natural question to determine the computational complexity of finding an optimal play,” say Michael Lampis at the KTH Royal Institute of Technology in Sweden and a few pals. 

The same question has been successfully asked of many board games, such as chess, Go and Othello, which tend to be PSPACE or EXPTIME-complete. But Scrabble is trickier because players do not know the order in which the tiles will be drawn meaning that chance plays a greater role.

The question that Lampis and co attempt to answer is this: given a Scrabble position how hard is it to determine the best playing strategy? 

They point out that in any given round, a Scrabble player is confronted with two tasks: deciding which word to form and deciding where to place it on the board. These tasks are related because the words that can be formed depend on the position of letters on the board. 

But which of these task is it that makes Scrabble hard? What Lampis and co show is that both are hard and give a proofs of each to back up their claim. That’s impressive because it allow us to ‘see’ why Scrabble is hard. 

“We establish that during the course of a game, Scrabble players need to perform not one, but two computationally hard tasks, which is probably the reason why Scrabble is so much fun to play,” they write.

That’s not to say computational complexity theorists are done and dusted with Scrabble. Their next task is to discover whether there is a polynomial-time algorithm to determine the move that would maximize the score achieved in this round.

Now that would be handy! 

Ref: arxiv.org/abs/1201.5298: Scrabble is PSPACE-Complete