I like to think of the board as a closed system: every move effects everything else. Let’s look at a game and watch how the numbers change over the course of a few moves. Here is the game code if you want to view the entire thing at the Quoridor Freeboard (ReBPNKwglK4p8Gkm5WkJ0CvCzSZGKEZmgmRkqm8QuSQCYmJiQkBAICZGQC2+IAhqBhRERiAgRkYgIE).

This game started with a sidewall from White. Black responded with a horizontal wall attack. At this point White is down moves, but up in walls. With Black in open territory, let’s see what White can do to lengthen Black’s path.

White plays the best kind of wall. Black wants to move once space down, but after White’s wall he has to walk 5 spaces around to get to the same place. Minus the cost of White’s turn, this wall can be thought of as a +4 wall. What’s not as immediately apparent is how many moves this wall will add to White’s total moves to the goal.

Over the next 3 turns White strikes twice more, winding Black around and forcing him to take more and more moves. From here we can look at a quick table of the moves to goal over the last six turns.

We can see that over the course of six turns White managed to add 6 more moves to Black’s path while only adding one to his own. With the wall count even, both players begin to march towards one another. Before we move, let’s look again at these numbers.

Here we can see each player’s closest goal in red, and cutting the distance between player’s in half (24/2 = 12), the two spaces where they will meet (pink). Indeed, not to be lost is the fact that it is Black’s move, and the space between the player’s is even, so Black will arrive first and have the benefit of the jump. Could White have known this before his attack? It would have taken incredible foresight, but it is at least worth mentioning. Players of all skill levels can succumb to tunnel vision, only paying attention to the number of moves they are stacking against their opponent, while forgetting that some of these may stack against themselves as well.

In this scenario, what does White do? He can’t delay moving and try to gain the jump by playing a wall, because Black has at least as many walls to respond with. White loses the waiting game. What would you do? This board is an excellent puzzle for newer players, as it is obvious that White, although down one move, has the advantage and can win this game (try it out at the Quoridor Freeboard with this code **j51nX6rLuSxzYIRBJTwjZYQw**)

Unfortunately White makes a mistake a not only walls first, but blocks himself in at the same time.

This gives Black the opportunity to close the path with one wall. Once the path is shared, the opponents trade the rest of their walls, White surrenders the jump, and Black’s victory is inevitable.

Screenshots from the Quoridor Freeboard, the most amazing thing to happen to Quoridor recently. Can you figure out how White should have won this game? Head over to the freeboard and record your solution. Post your answer in the comments below.

]]>After your oponent’s first step forward, place a vertical wall next to your opponent’s pawn.

Notice the wall does not block the back of your opponent’s board. Here is where this opening will usually lead to when used against opponents unfamiliar with the play.

Black’s move, and White is down 5 walls to 10. That said, White is 3 moves across the board and has Black boxed in. Black will tend to move towards the right side of the board. White can respond by building a horizontal wall across the board that will push Black back to the left.

White is spending walls but also White is managing steady progress across the board. On top of this, White is in full control of the movement flow. How did this happen? The initial sidewall is so immediate that when Black sees it, it “feels” like something must be done. At the same time. The sidewall pushes Black around, but it doesn’t create obstacles for White. Notice how the two available paths for White don’t wind or have extra spaces. The path to the right is shared, so it won’t matter if Black tries to elongate it. The path to the left is wide open, but White can close it quickly, or, if Black closes the right, White can “turn” the horizontal wall upwards and push the path (now shared) towards the goal.

Here is a great response. After White’s sidewall, Black plays a horizontal wall to begin closing the left side of the board.

With the left side of the board one wall from closing, Black can move to the right and quickly force his own path if White threatens to close the right side. Black also now has more immediate control over White’s path, and can use walls to attack. Because Black can quickly control his own path, White’s walls suddenly have less impact, and threaten to become an obstacle to White himself.

This is a great opening for intermediate players. It throws beginners off balance when they see it, and usually you can tie the game up early because the responses are unintuitive.

That said, this opening can be used all the way up to the highest levels, because you have to manage the entire board after opening moves.

Aggressive and complicated. This opening deserves much more in depth study.

Screenshots from the Quoridor Freeboard, the most amazing thing to happen to Quoridor recently. Paste this code (**QNBPNKwgKuL3AA) **into the textbox below the Quoridor Freeboard and click “input”. You will be able to walk through this opening and play around with different ideas. Paste any of the other codes below the images into the board to start from that position. I’m working on have the board embedded into this site.

The reason why beginners lose easily is because they keep attacking and fail to defend, which means that they cannot adapt to any change. If you look at the good openings, the one thing they have in common is that they serve a **defensive** function. Putting a fence right in front of your opponent at the start can seem like a good move but it really isn’t. Here is an example, with purple, the beginner, going downwards:

Purple can easily be corralled towards the closed side and turned around, and is running out of good moves fast.

If there’s one thing I would say to beginners, it’s to be patient and play more defensively early on in the game. It’s very easy to deflect the “Fool’s Mate” using the Gap. And that gives you a lot more options down the line, as well.

]]>**How to Perform It**

1. e8 e2

2. e7 e3

3. e6 e4

4. d5v

If done correctly, the game board should look like this:

**Application**

This can give you some control over the board and can let you somewhat control where the opponent is going to go.Some of these rely on rotations to stall them, meaning that it will consist of this:

This puts your opponent in a difficult position because both sides can be closed very easily, meaning that they must commit to either side and go back around the board. As they commit to one side, you can set a trap by blocking a little early to ensure yourself a path to the other side of the board with the least resistance. If done correctly, it may look similar to this scenario:

**Counters**

This opening is relatively new, so not many safe counters have not been found yet.

All in all, this way to open can easily be one of the best ways to aggressively play defensively.

]]>First, to restate the original question. I asked how many possible ways there are to put the 20 walls on the Quoridor board. For this question I’m ignoring illegal boards that block pawns or block paths to the goal. My motivation for asking the question was to whittle down Merten’s original estimate of how many total board positions there are in Quoridor.

Our answer starts by looking at the board and focusing on the vertices where walls are placed. If we narrow our view to one single row of the board, we can see one row contains 8 vertices.

What’s more is that each single vertex can take one of three values. It can be empty (no walls), it can have a horizontal wall, or it can have a vertical wall.

Along with this are some simple rules we have to follow. First, in one horizontal row there can’t be two “H vertices” next to each other. This would mean we have overlapping walls.

The same rule will apply for vertical walls, but for now we are dealing with one row so we can stick with just our horizontal wall rule.

From here calculate the total number of ways we can arrange the 8 vertices in one row. For example we could have a row that is completely blank, a row with a few horizontal walls, or a row with 8 vertical walls.

Because there are 3 possibilities for every vertex (X, V, H), we raise 3^8 to estimate the total number of rows. This gives us 6561, but we have to subtract from this number any rows that include two Hs next to each other. I’m not sure how to do this mathematically, but you can imagine a computer program that can check all 6561 rows quickly and eliminate any that violate our rule. This leaves us with 3344 ways to build one row without placing 2 Hs next to each other.

Next we stack. With our 3344 possible horizontal rows, we can simply stack them on top of each other. This time we have to add our rule for vertical walls. If a given row includes a V vertex, then the row above it or below it may not have a V in the same column.

8 rows stacked makes one board. Again, I’m not sure about the math here, but I can imagine a computer program doing two things. First it can find all of the permutations of our various rows stacked on top of each other in sets of 8, or, in other words, all possible boards. Then it can go through this list of boards and eliminate any that violate our rules about two V vertices in the same column.

This leaves us with one final check. Following our logic so far we could have a board of entirely vertical walls. This wouldn’t violate any of our previous rules, but it would violate the limit of 20 walls in a game of Quoridor. To solve this again we have to imagine a computer program checking each board.

Starting with the top row of a given board the computer would count the walls in that row. Moving on to the next row the computer would count the walls in that row and keep a running total. If we get over 20 then we know it is an illegal board and we can eliminate from our final set.

So the answer???

If you haven’t checked out the answer posted on MathSE I strongly recommend it. I think the author does a great job of explaining it, and he even took the time to answer my follow up questions. Thanks!

]]>You can see the situation at the end of the opening in the diagram to the right. Green has 7 fences left and Pink has 4 fences left.

The move count is: Green 9, Pink 7

Green has six Shiller paths:

7-2, 7-6-3, 7-6-4

8-6-2, 8-3, 8-4

Pink has six Shiller paths:

1-2-7, 1-2-6-8

3-6-7, 3-8

5-4-8, 5-4-6-7

The game is still pretty even at this point, despite the fence difference. Things are still pretty wide open. So it seems that Pink’s efforts did not yield much results at all.

Just to clarify: when I write Shiller path, what I mean is a path which goes through a distinct set of gaps (1-gaps and/or 2-gaps). This can probably be used interchangeably with path, but I just want to be specific about what I am counting. What interests us specifically in game analysis are the paths that can be manipulated by one or both opponents.

At this point, Green has closed the right side of the board to stop Pink from reorienting itself. Green has 5 fences left and Pink has 4 fences left.

The move count is: Green 10, Pink 12

Green has three Shiller paths:

7-2-1

5-3, 5-3-6

Pink has three Shiller paths:

7

2-1-6-3, 2-1-4-3

Both sides have obvious moves at this point. Green’s obvious move is to close 5 and reduce its path count to one, without affecting Pink’s (closing 7 would be ideal but it is a 2-gap, and at this point Green would never have the time to close it). Pink’s obvious move is to close 1 or 2, reducing its path count to one and keeping Green’s at two. This is what Pink does on its next move.

You can easily see how we’ve ended the midgame quite abruptly: there are no fenceable gaps left. This one fence by Pink has virtually wiped all the resources off the board, except for the branching on the top right due to h6v, which is easily repaired by Green due to its fence advantage.

The move count is now Green 11, Pink 12, so this is still anyone’s game.

]]>It is obvious that Quoridor has something to do with connection games, but it is not a paradigmatic connection game where one forms a chain between one side of the board to the other before one’s opponent. In fact, it seems at first glance to be the exact opposite of that: a game where you try to prevent your opponent from forming a connection. But of course this is the wrong way of looking at things, since the rules explicitly forbid you from doing this (and that would be a much less interesting game!).

Finally I got the key idea from mulling about the Shiller Principle. I realized that the “paths to goal” count is a more generalized concept than actual concrete paths. At the beginning of the game, one may say that the number of possible concrete paths one can take to the goal is extremely large (I won’t say infinite, if only playing a game of infinite length is, well, impossible).

And if we start from that premise, then one can further theorize that laying down fences lowers the number of actual paths, and that the mechanism by which one wins is by whittling options down for your opponent, and whittling away the most attractive options, leaving them with fewer, and worse, options. So in essence it seems that the “reverse” view is correct, insofar as you start with near-infinite connections and try to lower that number, instead of starting with no connections and trying to form one.

Of course, no idea is entirely new, and someone already understood that before me:

[T]he number of ways for a player to reach his target edge decreases as more and more fences are placed on the board: Quoridor begins in a maximal state of connective potential that converges as the game progresses.

Cameron Browne, inConnection Games: Variations on a Theme

But I don’t find this perspective to be the most useful in terms of understanding how this game works. I want to suggest a different, and weirder, idea: Quoridor as a resource management game.

Going back to the Shiller Principle, it is true that at the beginning there is only one optimal path (or zero, depending on whether you started first). As the game develops, there may be a number of Shiller paths possible, but the main reason why it matters is because of fenceable gaps (i.e. because those paths can be closed down).

My idea is that, perhaps not explicitly but as part of strategy, fenceable gaps are a resource that is generated and spent during play. Depending where the gap is, it can be a resource or a liability to either player, and part of the unpredictability of this game is that one can easily be turned into the other.

From this viewpoint, any average game presents an abundance of riches. On the diagram to the right, fenceable gaps are numbered in red, and gaps fenceable in two are numbered in dark yellow.

Some of these are more of a resource for one side than the other. Pink’s direct line to the goal involves either 1 or 3-6, while Green’s direct line to the goal involves 2-1-7-10, 6-9 or 6-8-5. On the whole, we can see that Pink has a big advantage because its paths involve fewer fenceable gaps, which means the opponent will run out of resources much faster. As a result, Pink also has only six Schiller paths while Green has ten.

Other fenceable gaps are resources in that they force moves to the opponent: Pink here understandably closes down 4 on its next move, which drops his Shiller path count to a mere two, forcing Green to close down 1 as it is the least defensible path.

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