Quoridor and Chess are Inverted.

Here is a thing chess cannot do. Take any Quoridor position, at any moment, and I can tell you precisely who is ahead. Not a guess, not a heuristic, not a number squeezed out of a neural network trained on millions of games. The actual answer.

Count the shortest path from your pawn to your goal row. Count your opponent's. Whoever needs fewer steps, and it is their turn, is winning. It is a breadth first search on an 81 square grid and it takes under a microsecond.

So the game should be easy. It is not. And the reason it is not turns out to say something interesting about what actually makes a game hard.

Chess and Quoridor are inverted problems

Chess is difficult to evaluate and comparatively narrow to search. A typical chess position offers around 35 legal moves, but telling a good position from a bad one is so hard that it took the field seventy years and eventually a neural network to do it well. The tree is manageable. The judgment is brutal.

Quoridor is the exact opposite.

The judgment is free. The tree is enormous. From the opening position you have three pawn moves, because your pawn is on the back rank and can only go forward or sideways. You also have 128 legal wall placements. Every one of the 64 intersections on the board takes a wall horizontally or vertically, and at the start every one of them is legal.

That number does not stay at 131, and how it falls is the shape of the game. Every wall placed is a wall gone forever, and the branching factor drains away with them.

Three legal moves

That last sentence is worth sitting with. When the walls run out, a Quoridor position has three legal moves and the entire game reduces to comparing two numbers you already knew. There is nothing left to calculate. It is not a hard endgame or a subtle one. It is arithmetic.

Chess does not do this. A chess endgame with four pieces is still genuinely difficult and people have written books about single ones. Quoridor has a phase transition. At some point in the middle of the game the position stops being a puzzle and becomes a receipt.

Which means the whole game, all of it, lives in the window before that. Every decision that matters is made while walls are still in hand.

You get 128 options and 20 decisions

There are 128 places a wall can go. Between the two players there are only 20 walls. So at most 20 of those 128 possibilities will ever happen, and you have to think about all 128 anyway, every single turn, because the one you want might be any of them.

And they are permanent. This is the part that separates Quoridor from almost every other abstract strategy game. A chess piece that goes to a bad square can come back. A Quoridor wall is placed once and stands for the rest of the game. Twenty irreversible decisions, spread across a game that lasts about fifty turns, each one chosen from a hundred and something candidates.

There is a rule that makes this much worse, and it is the best rule in the game: you may never place a wall that leaves either player with no path to their goal. You can slow someone down as much as you like. You can never seal them in.

It is the rule that stops the game from being a stalemate simulator. It is also expensive. To know whether a wall is legal you cannot just look at the board, you have to place it and run a pathfinding search to check both players can still get home. Do that for all 128 candidates and the cost adds up.

What this does to engines

We recently had all eight of our bots play a round robin against each other, 112 games, and got a result that looked like a bug.

Our Hard bot lost to our Medium bot. Not narrowly. Medium won their head to head three games to one and won twice as often across the whole tournament.

The difference between them is not what you would guess. Hard searches one move deeper than Medium. But to afford that depth it only considers twelve candidate wall placements per turn, where Medium considers every legal one. Hard sees further along a much narrower path.

In chess that trade is usually fine. Most chess moves are junk and a decent engine can throw them out cheaply. In Quoridor the walls are the game, and cutting the list to twelve throws away the move you needed. Depth bought with breadth is a bad deal here, and the game punished it by about twenty percentage points of win rate.

A second result pointed the same direction. Our strongest neural engine looks roughly fourteen moves ahead. Another engine, looking nowhere near that far, beat it three games to one. Small sample, and we are not treating it as settled. But nothing in these numbers rewards depth for its own sake.

What it means if you are the one playing

The practical version of all this is short.

Counting the race is the easy part, so do it every turn and stop treating it like skill. It is bookkeeping. Two numbers, yours and theirs, and who is on move.

The skill is knowing what those numbers will look like after the walls land. A wall that adds two steps to your opponent's path and costs you a turn has broken even at best. A wall that adds six, or that forces them into a corridor your next wall can close, is the whole game. Since you only get ten of them and they never come back, the question is never "is this a good wall" but "is this the best use of a wall I will never have again."

And when the last wall goes down, stop thinking. Count. You already have the answer.