Wall or Advance: The Decision You Make Every Turn

Every turn in Quoridor asks the same question, and it is the only question the game ever asks: step, or build. You cannot do both in one turn and you cannot pass, so across a full game you will answer it a few dozen times. Games are rarely lost to one catastrophic move. They are lost to a dozen of these answered by feel, each slightly wrong in the same direction.

The choice feels hard because the options look like they are measured in different units — distance for a step, obstruction for a wall. They are not. Both are tempo, and once you can convert between them the decision collapses into a checklist you can run in a few seconds.

Positions below use columns a through i left to right, and rows 1 through 9 from your own baseline to your goal edge. Your opponent starts on row 9 and travels toward row 1.

The threshold, borrowed

Two results from elsewhere in this series set up everything below. Take them as given here rather than deriving them again.

The number that decides the game is the difference between your shortest route and your opponent's, with the turn folded in; Tempo and the Race covers how to produce those two numbers. And Wall Economy prices the wall itself: what a wall returns is the squares it adds to their route, minus the squares it adds to yours, minus one for the turn you spent not walking.

Name the two quantities. k is the squares a wall adds to their path, j the squares it adds to yours — both are in the Glossary. Where that pricing lands is a bar, and the bar is the one thing this page borrows outright: run that expression down its ladder and a k − j of 2 is the floor for considering a wall at all, and 3 is the number that means yes.

What those two pages leave open is the question you actually face on your turn: given the threshold, how do you find out whether such a wall exists at all, without recounting the whole board? Usually it does not exist. Knowing why, and knowing it in a few seconds, is most of the skill.

Step one: count first, and let the count gate the search

Counting is cheap. Searching the board for a good wall is expensive — an empty nine-by-nine board offers 128 legal wall placements, and reading any one of them properly costs you a recount of your opponent's route. So do the cheap thing first, and use it to decide whether the expensive thing is worth doing at all.

If you are ahead and it is your move, the default is to walk, and most of the time you can skip the search entirely. A step preserves a race you are already winning; a wall risks converting it into a wall-trading contest you did not need; and margin above the winning threshold buys nothing anyway, which Endgames turns into an exact allowance of free turns. Meanwhile the wall you did not place goes on appreciating as the board narrows (Wall Economy prices that curve).

The gate is at its strongest in the opening, where the count is dead level and, for a pawn walking up the middle of an empty board, no single wall clears the bar — which is why the mainline is simply to advance, and why Openings has to work to find the early walls that are exceptions. The qualifier about the middle of the board is doing real work there; step two is where it earns its keep.

The gate has one override. Search anyway when something on the board is visibly loaded — an opponent pawn on or beside an edge column, a wall already standing near their route, a pawn that has committed itself to a corridor. Those are the conditions under which the threshold becomes reachable at all.

Step two: does a wall that clears the threshold even exist?

Prune before you search. Only a wall that touches your opponent's current shortest route can raise k above zero, because a wall anywhere else leaves that route intact and therefore changes nothing — and if they hold two routes of equal length, one wall has to touch both. While that route is a straight descent, only walls lying across it can touch it at all: a wall running parallel to a pawn's column does not obstruct the pawn, it only blocks sideways movement, which matters later, once the detour has to cross it. Those two filters cut a hundred-plus placements down to a handful.

Now the arithmetic. Suppose both pawns have walked three squares straight up the middle: yours on e4 needing five more, theirs on e6 needing five more, board otherwise empty, your move. The obvious wall is a horizontal one in the groove between rows 5 and 6 covering columns d and e, laid directly across their front. Recount their route: e6 to f6, f6 to f5, then straight down — six moves where five used to do. k = 1. It matches a step and costs a wall. Walk instead.

The reason is structural, not particular to that square. A wall spans exactly two squares, so it covers the pawn's own column and one neighbour — and a pawn standing in open space has two shoulders. The other shoulder is still open, the detour is one sidestep and a resumption, and so on an open board the second-best route is almost always exactly one move longer than the best. k = 1 is the ceiling for a lone wall in open space, no matter how well aimed.

Now change one thing. Their pawn is on a6 instead of e6, still five from goal, and the same kind of wall goes into the rows 5/6 groove covering columns a and b. The pawn now has one shoulder. To reach row 5 it has to get to column c: a6 to b6 to c6 to c5, three moves where one used to do. k = 2. The wall did not improve; the board did. The edge supplied the second block for free.

So the search rule is not really about walls, it is about shoulders. A wall clears the threshold only when it covers the route and every bypass beside it, and two squares of wall can manage that only with help — help being whatever supplies the missing shoulder: a board edge, or a wall already standing.

Note what that does not say. It is not a rule about proximity to the edge. A pawn one column in from the rail with the lane on its far side already sealed has exactly one shoulder, the same as a pawn on the rail — but only one of the two walls covering its column also covers that shoulder, and its mirror image leaves the rail-side lane open and is worth a mere k = 1, where it can be placed at all. Count the surviving bypasses, not the distance to the rail; the worked position at the end of this page turns on that distinction, and on that qualifier.

The help need not be your own, either. Once a wall is placed it is terrain, and terrain belongs to nobody. The deliberate version, where one wall's detour is aimed straight into the next, is the foundation of Traps and Cages.

Step three: what does the wall cost your own route?

k is only half the figure. A wall lengthens your path too whenever your route crosses the same ground — that is j, and it comes off at full weight, with no discount for the wall being yours. So the operational form is a second count: before placing, recount your own shortest route with the hypothetical wall on the board, as carefully as you counted theirs. Walls in the middle of the board routinely carry a nonzero j, because your own route usually runs somewhere through there. Walls deep in your opponent's half rarely do.

j also hides in the future. Your route today is not the route you will be walking after they answer, and a wall that costs you nothing on the current line can cost you two once your line shifts a column — which is exactly what a wall of theirs is designed to make happen. Worse, your wall can become the anchor that makes their next wall worth two against you. Before you place, read the wall you are about to build from your opponent's side of the board and ask what it is worth to them; if the answer is "a free shoulder", you are building half their structure for them.

One tiebreak for when the arithmetic ties. Placements are exclusive, so a wall you place is one they can never place there. When two candidates score the same k − j, take the groove you would hate to see used against you — Wall Economy counts that exclusivity on the cost side, where it belongs.

Step four: supply, measured against theirs

The question is not how many walls you hold. It is how many you hold compared to what they can answer with. Running out first carries no penalty by rule; it simply leaves you choosing between one move a turn while they choose between two (Wall Economy develops the supply argument).

Say you are two behind, you hold two walls, and they hold five. Each k − j = 2 wall you find gains one tempo over stepping, so two of them erase the gap exactly — and then they still have five, each of which can put a tempo straight back. You cannot win a trading war down two to five.

That arithmetic assumes each of those five walls can actually be cashed, and a held wall only prices at a tempo where a qualifying placement exists for it — Wall Economy sets out which conditions supply one. What step four adds is that you have to run that reading twice, once per side, and the two readings almost never agree, because the ground in front of your pawn and the ground in front of theirs are different ground. Five walls aimed at a route crossing open middle board are five turns that cannot yet clear the threshold; your two aimed at a pawn already committed to a file may both be live this turn. Compare cashable placements in both directions, and take the reading for the turn you mean to spend on rather than for today.

Down two to five, then, a wall has to do something a step cannot: it must be aimed at a structural weakness yielding a k well above two, or at the one groove that denies their best future placement. If neither is there, walk, hold the walls, and wait for them to commit to a route worth attacking.

There is a hard boundary underneath all of this. When both supplies reach zero the maze never changes again and the position is a pure race — see Endgames. The moment you spend your last wall you should already know how that race comes out.

Step five: commit, then recount

Placing the wall does not end the decision. Their reply is chosen specifically to devalue it, and often the reroute is a move they were content to make anyway. Recount on your very next turn instead of trusting the number you computed before placing.

The reason the number can move so far is that wall value is not additive, in either direction. Two walls can be worth more than the sum of their parts when the second closes the detour the first one opened; that superlinear case is the whole of Traps and Cages. The same non-additivity runs the other way too, and that direction is what this step guards against. Wall their column and they step one file left; aim your second wall at the lane they have already abandoned and its k is zero, because your first wall already spent it. Two walls, one detour, one tempo.

So run the test rather than estimating: recount after the placement, and if the number did not move, you did not place a wall, you discarded one. The corollary is a placement preference — walls near the live point of the position are less likely to be made redundant by your own later play, because the pawn has not yet had a chance to route around the ground they cover.

Running the whole thing on one position

Your pawn is on e5, four from your goal. Their pawn is on b4, three from theirs. Earlier they placed a vertical wall in the groove between columns b and c spanning rows 3 and 4 — a wall meant to stop your pawn crossing into the left-hand files back when you might have gone that way. You never did, and it has been dead terrain ever since. They have one wall left; you have four. It is your move.

Step one. You need four moves, they need three, and you move first, so they still finish first. You are one tempo behind. The search is mandatory.

Step two. Their pawn is on column b — beside the edge, not on it — so on an empty board neither wall covering its column would reach k = 2: cover a and b and they sidestep to c, cover b and c and they sidestep to a, one move either way. But the board is not empty, and their own vertical wall sealing b from c across rows 3 and 4 changes both candidates — in two different ways.

Cover a and b, a horizontal wall laid in the groove between rows 3 and 4, and that pawn has a3 walled, b3 walled, and c4 sealed off by their own earlier wall. It has to retreat to b5, cross to c5, and come back down the c column: six moves to goal instead of three. k = 3. The edge did not do that; the standing wall supplied the missing shoulder — step two paying out as advertised.

Now reach for the mirror, the horizontal wall covering b and c in that same groove, and there is nothing to reach for. The b/c groove and the 3/4 groove meet at a single junction; their vertical wall is already pinned to it, and the horizontal wall you want would be pinned to the same junction. Two wall pieces cannot occupy one crossing point, so that placement is not legal at all — the prohibition is in The Complete Rules, and the engine enforces it by rejecting any wall sharing an anchor point with one already standing, whatever its orientation.

That is the second lesson, and nothing on the board announces it. A standing wall deletes placements as well as obstructing routes — Wall Economy prices that exclusivity in full. The help step two sends you looking for can have already consumed the option you wanted it for. Here it has not — a and b survives and is worth three — but the reflex that says if that wall works, its mirror must be worth something too is not merely optimistic about the value. Sometimes, as here, there is no move there to value.

Step three. The wall sits in the bottom-left, behind you and three files off your column, and your own route up the e file never touches it. j = 0, so k − j = 3.

Step four. They hold one wall. With their pawn buried in the corner and the centre otherwise empty, the placements that would reach k − j = 2 against your route are not on the board, so that wall buys back one tempo at the very most. After your wall you need four and they need six, with them to move — a two-tempo lead. One answering wall does not overturn it. Green light.

Step five. Place it, then recount after their reply. If they abandon the descent and start building against you instead, the race number is the only thing that will tell you whether you can still afford to keep walking.

The procedure is training wheels, not a permanent ritual. After a few dozen games, steps one and two take a second between them, and you only descend into the rest on the rare turns when step two comes back with something. The habit worth keeping forever is the smaller one inside it: never place a wall you have not counted from both sides of the board. The rest of the series branches from Strategy.