# Why is it better to control a vertical/horizontal than diagonal?

I'm trying to truly understand why Rooks are more valuable than Bishops. Yes, they control more squares. But why?

I understand that a rook forms a line that pieces cannot pass through, and this is related to the fact that it controls squares of multiple colors. A diagonal does not block a piece in the same way.

But I just cannot wrap my head around why controlling all squares in a cardinal direction is any better than controlling all squares in an ordinal direction. Does anybody have a mathematical or logical explanation for this? Why does controlling squares cardinally result in more total squares controlled than doing so ordinally?

• You "understand that a rook forms a line that pieces cannot pass through". Do you understand that, specifically, a rook guards all squares on a line that the enemy king cannot pass through, and can thus confine that king? This is part of the reason why K+R can mate a lone king though K+B cannot. Jul 5, 2023 at 8:54
• Its due to the blocking power that Rooks have v Bishops. To create a barrier with Bishops you require a pair of them. Where as a single Rook / Queen can effectively slice the board in 2 ,3, 4 Jul 5, 2023 at 13:03
• So you don't understand why a fence with a bunch of holes in it is worse than a fence with no holes in it? Jul 5, 2023 at 17:24
• @Kevin no, the question I'm asking is why a diagonal fence has holes but a straight one does not
– Alec
Jul 5, 2023 at 22:25
• "Does anybody have a mathematical or logical explanation for this? Why does controlling squares cardinally result in more total squares controlled than doing so ordinally?" - did you try putting a bishop on an open chess board, marking which squares it can reach in a single move, and counting them? Then doing the same for a rook, and comparing? I don't understand how there is a question. Jul 7, 2023 at 1:34

I'm not sure exactly what question you're asking, but here are a few ideas related to why rooks might be stronger than bishops:

• Bishops are easier to block with pawns. If a bishop sits on a nice open diagonal, it's quite likely that the opponent can adjust their pawn structure to block it. On the other hand, if a rook sits on a nice open file, then mostly the file stays open.
• On an empty board, from any square a rook covers 14 other squares. A bishop covers somewhere between 7 and 13. Put another way, some diagonals are very short, while all ranks and files are long.
• As you mention: although a bishop can cover quite a few squares at a time, there are half the squares on the board that it can never attack. The rook is more flexible and there is no square it can't attack. For example, on an open board it's much easier for a rook to harass a bishop than for a bishop to harass a rook (although in a more cramped position, a bishop may be able to keep a rook off particular important squares).
• Rook and king alone can mate, unlike bishop and king. Similarly, a rook can deliver back-rank mates.
• A rook can attack a pawn in such a way that the pawn can't move out of the attack. A bishop can't (unless the pawn is blocked).
• These points are nice. I understand that attacking squares horizontally& vertically is stronger as you get to control 14 squares rather than 7 - 13. What I don't understand is why that's the case. Why does controlling horizontal/vertical yield more squares covered than diagonal?
– Alec
Jul 4, 2023 at 23:54
• @Alec the reason that is the case is simply due to the geometry of the board. Across the entire board, there is no variation in the length of any rank or file, so a rook will always be able to see 14 other squares on an empty board. However, some diagonals are much longer than others, and some squares only see a single diagonal. Look at the board and note how many squares a bishop can see on different squares. E.g. a bishop on a1 can only see 7 squares, but a bishop on d4 can see 13. Jul 5, 2023 at 0:57
• @Alec I think I see what you're asking. Mathematically, you could say that on an n-by-n board, the rank of a square (x,y) always has length y + (n-y) = n. If you can't go far right, then compensatingly you can go a long way left. But its NE/SW diagonal has length min{x,y}+min{n-x, n-y}, which is at most n (precisely when x and y are the same), and will be very small if x and y are very different. For example, if you're in the SE corner (large x, small y), then x being large stops you going far NE, but there is no compensation because simultaneously y being small stops you going far SW. Jul 5, 2023 at 7:36
• Note that, on a cylindrical chessboard, a bishop placed anywhere on an empty board will control 14 squares, the same as a rook. I don't know if that makes the bishop equal to a rook in cylindrical chess; I've never played that variant.
– bof
Jul 5, 2023 at 22:29
• @bof nice point! although weirdly, it seems to be 13, one less than a rook. The two diagonals that the bishop controls intersect in one square (in addition to the one on which the bishop sits). Jul 5, 2023 at 22:53

I am answering the mathematically-oriented question why are rooks controlling more squares (the chess question of why are rooks better pieces is well-answered here: Why is the rook worth 5 points while the bishop is only worth 3?)

Rook on position (X,Y) can move to the left: X-1 + top: Y-1 + right: 8-X + bottom: 8-Y. Sum this up and you get 14 tiles it can move to. No matter where the rook is. On a different rectangular board you get N and M instead of 8s, leading to N+M-2 squares it can move to. This is because movement in one direction is independent on the other direction and all ranks and files are as long.

Take bishop on the same position (X,Y). Going left-up, it can move by min(X-1, Y-1). Likewise for other directions. Those minimums are because bishop reaches the end of either row or column first and cannot move further - limiting bishop's mobility. In the optimal case for bishop in the center of an odd-sized square board, it can move to as many spots as a rook can (2N-2); however, if the bishop is in a corner, it can only move to N-1 squares.

But you can construct a board (diamond-shaped) where number of moves for bishop is constant while number of possible rook squares changes (but is as high or higher than number of bishop moves). You can also make boards where a rook reaches fewer squares than a bishop does in a single move (eg a tri-diagonal strip starting on A1 and go up to H8 together with both side diagonals) - so, even though rook reaches all tiles, bishop moves faster over that board.

• The mathematical arguments, and the generalisation in the last paragraph is brilliant. Thank you 🙏🏼 Jul 5, 2023 at 13:05
• The relative value of the two pieces on the diamond-shaped board intrigues me. The bishop's relative mobility is increased, but it still only has access to half the board and can't cut off an enemy king. Jul 5, 2023 at 16:01
• Exactly what I was looking for. The reason that a Rook controls more squares is because the board itself is straight. If it were oriented diagonally, then the Bishop would control more squares and the # controlled would be invariant instead of the other way around. Makes perfect sense.
– Alec
Jul 5, 2023 at 22:27
• @Alec Actually on diamond shaped board (imagine a small one with a3, b2-4, c1-5, d2-4 e3) rook controls as many or more squares. It is just that bishop always controls 4 while rook in a corner gets to 4 and rook in the center has 8. You need diagonal board for that - extend that diamond shaped one along one diagonal far enough and you get to bishop reaching more squares than a rook in a single move. (while the worst case for bishop is a very long and narrow rectangular board). Jul 5, 2023 at 22:37

Apart from the obvious advantages of the rook that you also mentioned (according to the rules of chess) I think your question boils down to this:

Why is the bishop vs. rook on a chessboard different from vertical + horizontal lines vs. diagonal directions in an open (infinite) plane?

I'm trying to answer this here, attempting to clarify exactly where those advantages come from (mathematically? geometrically?).

## Bounds of the board

The chess board is a square board, not an infinite plane, so the directions matter. The rook has an advantage because the directions it can move in coincide with the square's sides.

The rook cannot be stuck in a corner, a bishop can, so that one of the directions is blocked and it can only move in one direction. This means that while both pieces can reach any of their possible destinations in 2 moves, the rook can decide which of the 2 moves it takes first, but a 'stuck' bishop in a corner can't.

## Grid

The chessboard is not just a square, but a grid of squares (so you could say it is discrete). If you think about it, the previous point about the bounds of the board holds even if there is no grid (if any piece could move an arbitrarily small distance), but in addition, the bishop also loses half of the board, and that is a consequence of the grid.

• Beautifully explained, Thank you 🙏🏼. The bottom two diagrams are brilliant. Jul 5, 2023 at 13:02
• "The rook has an advantage because the directions it can move in coincide with the square's sides." - perfect answer
– Alec
Jul 5, 2023 at 22:33

But I just cannot wrap my head around why controlling all squares in a cardinal direction is any better than controlling all squares in an ordinal direction.

We can picture a world where they actually are the same. Imagine playing chess on a board from the game of Hex. Here, a king can only move to the six adjacent tiles (green). A "rank" and a "diagonal" are the same. The rook and bishop both block the king.

But now we change the rules to let the king jump two squares up and to the left, or two squares down and to the right. Just like in chess, the king has 8 squares to go to. And just like in chess, the poor bishop can be bypassed.

So this is how we get a chess board: we squish the Hex board into a square, and we say that a tile is adjacent to the tile that is up+left of it, as well as the tile down+right of it. We make a new diagonal for the up+left and down+right direction, pictured below in pink. It has the same "bishop" problem as the opposite diagonal: the king can hop over it.

Two key points:

1. Because of where we added the new green dots, a rook can control up to three squares adjacent to the king, but a bishop can only ever control two.
2. On a chessboard, the king can only move one over by one rank or file at a time, but can move two diagonals at a time. (On a chessboard, there are only 8 ranks and 8 files, but there are 15 diagonals that run up-left and 15 diagonals that run up-right.)

The other answers here are great. Inspired by them, I came up with a mathematical reason.

Basically, diagonal walls have holes because bishops are limited to half the squares.

Define a move as a vector: `<+xpos, +ypos>`.

Any Rook move has the following vector basis: `{<1,0>, <0,1>, <-1,0>, <0,-1>}`

Any Bishop has the following vector basis: `{<1,1>, <-1,-1> <-1,1>, <1,-1>}`

When a Bishop moves, `sum(Δx, Δy)` must be even. Therefore, it must move to a square of the same color. When a rook moves, `sum(Δx, Δy)` may be even or odd, so it can move to a square of either color.

As an extension, for a knight, `sum(Δx, Δy)` must be odd. Therefore, it must move to a square of a different color.

The reason that bishops can only reach half the board is because they are limited to moves over an even amount of squares, while rooks can move over an even or an odd amount of squares.

This means that bishops lack the same blocking capabilities as rooks. Also, they see fewer squares (7-13 vs 14), for the same reason.

A chessboard is a finite, upright square lattice of size n=8. A rook can move to any position a(1,0) + b(0,1) but a dark bishop can only move to c(1,1) + d(1,-1), where a,b,c,d are any integers that keep the resulting position within the board. Consider for example the position (1,0), which is unreachable by the dark bishop.

This is because the rook's vectors span the upright square lattice, but the bishop's vectors only span half the space.

I think major parts of this is are actually the interactions with other pieces. In particular, one's own pawns and the opponent's king. A rook can create an uncrossable barrier for a king whereas a bishop cannot. And since pawns advance along the file, the rook can protect them their entire journey towards as they advance towards promotion, without the rook having to move.

Additionally, the rook always attacks its maximum number of squares - an entire rank and file. Whereas the bishop's attack power is only maximised when it's in the centre of the board. In any corner, it only attacks one diagonal.