# Why is the rook worth 5 points while the bishop is only worth 3?

Why is the rook worth 5 points while the bishop is only worth 3? Rooks only move straight, but bishops only move diagonally. Is there something that makes moving straight advantageous to moving diagonally?

• It is good to remember that 1) a Bishop will allways be bound to the color of diagonal it was born, 2) Losing a Bishop will make you ineffective on a color of diagonal, 3) Diagonals have different sizes accross the board while columns and rows are always the same size. A Bishop will have to conquer the center to have full effectiveness while the rook will have the same reach from any square of the board(not counting here threats or movement block imposed by opponents)
– user13443
Jun 2, 2017 at 17:30
• Note: One bishop is worth 3 points, but having two of them (on different colours, as usual) gives you an additional Pair bonus of 0.5 points. Jun 6, 2017 at 16:19

Have a look at the wikipedia article on relative chess piece values, it provides an extensive discussion on the matter.

To see how you can compare two given pieces, let's consider your bishop vs rook question at a basic level:

• A rook's movement is not restricted to a color, unlike the bishop's. This makes half of the board squares inaccessible to a bishop.
• Assuming an empty board, the rook can always see 14 other squares from any other square, whereas a bishop's scope of legal moves varies between 7 to 13 depending on the square, the latter only reached when the bishop is centralised, something the rook doesn't care about!
• A rook has an additional way of moving when it combines with the king to castle, the bishop doesn't.
• A king and rook are sufficient to checkmate, but bishop+king vs king is a draw, as unlike the rook, a bishop cannot cut-off the king's path as it can always escape via the opposite square color of the bishop. Additionally, this difference allows a rook to achieve stalemate setups that a bishop cannot, as the rook is able to continuously check the king no matter where it goes.
• When combined, a pair of rooks can create a battery along a file/row, but bishops can only create batteries with the queen, or other promoted bishops of same color but those are not always a given. But now we're getting into more advanced details, so let's stop here.

So you see the collection of these properties make it clear that in an individual comparison, a rook's worth ought to be more than the bishop's. Now try to perform similar comparisons between any other pair of pieces in order to convince yourself where their relative values stand. Remember that key basic properties are:

• Type of movement both individually and combined
• mobility and range
• ability to deliver checkmate when combined with a king
• square dependence (such as color, and centrality dependence of bishops)

You need to understand that the point system is only a rough guideline meant to assist you in evaluating positions or in deciding on potential exchanges. Many factors, particularly the pawn structure, influences how valuable pieces are. Rooks tend to be better in open positions with fewer pieces/pawns on the board, bishops can get hindered in closed positions by pawns of the same color ("bad bishop"), etc.

Despite of all this, the 5/3 values for rook/bishop (or the piece valuation system in general) is surprisingly reliable.

A rook is generally more valuable than a bishop because:

1. it can reach all squares of the board, while a bishop can stay only on squares of the same color
2. it can mate in KR vs K, while you cannot win KB vs K.
3. it covers more squares (can always move to 14 squares on an empty board independent on where it is placed, while with a bishop it can vary between 7 (corner) and 13 (center) squares)

The exact values "5" and "3" are basically found from experience and since there is a bit of a drawing margin in chess, the exact values don't really matter.

One notable and very common exception to the rule is found in many (beginner's) opening play where a bishop and knight can be exchanged for a rook + pawn on f2 or f7. In these cases almost always having two light pieces is better even though you exchange 6 pawn units for 6 pawn units.

• A nice addition would be the fact that rooks can double up, while bishops can't Jun 6, 2017 at 23:55
• @VinceEmigh That's misleading. Rooks can be redundant, while bishops can't. Jun 15, 2017 at 6:47
• @PartyLovingHermit Bishops can easily be redundant... Wanna elaborate? Jun 15, 2017 at 13:39
• Bishops of one side are of opposite color since underpromotion to a bishop is almost never done in games. Rooks however, must share the same ranks and files in many cases. That's why the side with two rooks tries to exchange one of them when going against a rook and minor piece(s). The exception is when they're attacking, especially when they're on the second/seventh rank (you are right in that sense.) Jun 16, 2017 at 2:28
• @PartyLovingHermit Your comment above makes absolutely zero sense: "Rooks however, must share the same ranks and files in many cases. That's why the side with two rooks tries to exchange one of them when going against a rook and minor piece" there is no "must" in chess, everything depends on the position, always. Mar 7, 2018 at 22:54

I would like to add another example.

1. A rook can isolate a lonely king from a portion of the board by forming a barrier. A bishop cannot; for that same task you need 2 bishops. That means, roughly, 1R~2B

2. 2 rooks can form a barrier that blocks a knight. 2 bishops cannot; for that same task you need 3 bishops. In this other scenario we have, roughly, 2R~3B

Averaging 1. and 2., we see that the quotient R/B lies between 1.5 and 2. Which seems to be in accordance to the theoretical values R~5P vs B~3P.

The pawn point system is only a very rough guideline to judge a piece's value. In the beginning of the game (as long as there are no (half) open lines for the Rooks to operate), a Bishop might very well be more valuable than a Rook. The Rooks get better and better the less pieces there are on the board, because they are more mobile "in vacuum".

Not only can they potentionally reach all squares (while a Bishop can only access either black or white ones, which sucks for mating a King in the wrong corner), they are also less restricted by the board's borders: No matter where a Rook is placed on an empty board, it can ALWAYS move to 14 squares (7 on its line, 7 on its rank). A Bishop in a corner can only move to the 7 squares of the diagonal to the other corner. Even in the best case (when placed in one of the 4 center squares), they have access to at most 13 squares. Thus, it's usually a good idea to keep your Rooks around for a longer time than your Bishops.

The point values are based primarily on how many squares a piece can command. Not only can rooks command more squares on an open board, but they can cover both colors, while bishops are restricted to squares of their own color. one on the black squares and the other on the white squares. Incidentally, both pieces move "straight", only the rooks do so horizontally or vertically while the bishops move diagonally.

You ask a basic but significant question. The question can be answered in either of two ways.

On the one hand, chess masters have found, roughly, that trading

• a rook and a pawn for
• two minor pieces (bishops and/or knights)

tends (averaged over many games) to leave realistic winning chances on both sides; whereas trading

• a rook for
• two minor pieces and a pawn

seldom leaves realistic winning chances on both sides unless a tactical combination is in view. By assigning the rook a value of 5 and the bishop a value of 3, the masters have given us a handy metric that roughly reflects a master's experience in the matter.

On the other hand, a rook can reach all squares on the board whereas a bishop can reach only half. A rook can easily hide from a bishop by the trivial expedient of occupying a square the bishop can never reach; whereas a bishop cannot hide from a rook in this way.

The two perspectives mesh. Either or both explains the valuation.

(By way of analogy, consider that, in contract bridge, experience finds that an ace almost always takes a trick, a king usually takes one, a queen sometimes takes one, a jack can take one but usually doesn't, and a ten seldom takes one. This is not hard to understand, since the ace is the highest card, but the rough metric bridge masters have developed from experience is to value an ace at 4 points, a king at 3, a queen at 2 and a jack at 1.)

From playing chess variants, we know that for Cylindrical Chess the strength of Rook and Bishop are practically equal. In this variant, both pieces have the same board mobility.

So, the main reason for the diminished strength of the Bishop in Classical Chess is its restricted mobility on the board: On an empty board a rook can always reach 14 squares, while a Bishop can reach between 13 (placed in the centre) and 7 (placed in a corner) squares.

Other factors (like the "can-mate" property or the "king-restriction" property) may play a role, too, but they are not decisive.

Im a bit surprised that nobody yet has mentioned that if you still possess both bishops they can in combination cover squares of both color. One Bishop plugs the holes that the other Bishop leaves. This is why it is often considered advantageous to have "the bishop pair".

But a great deal depends on the scope that the pieces have in the particular position, especially if there are weak pawns that may come under attack, perhaps after initial manoeuvres. Also the mobility of the King, if you are in, or about to be in, an endgame. The king and a bishop can attack a pawn, and the rook cannot defend that pawn by itself. If the King joins the defense, perhaps the other bishop can chase it off.

Counting points is never more than a start. I always do it automatically when evaluating an exchange, but then the immediate question has to be, "in this particular position, are there circumstances that might outweigh that judgement?"

Since Bishops only move diagonally, a Bishop can only go to a square of the same color as it started on. White Bishop can't go on black squares, and black Bishop can't go on white squares. However, one Rook has the possibility to access all squares on the chessboard. This is one reason why Rooks are worth more than Bishops.

Additionally, rooks can also control more of the board at once, especially on the edges. When in a corner, a Rook controls all the squares of a rank and a file. When a Bishop is in a corner, it's severely limited because its line of attack is only 7 squares diagonal.

Ellie points out that a rook, wherever it is, sees more squares than a bishop can. The reason for this is the shape of the board: rook-lines run parallel to the board's edges. If the board was of squares but was diamond-shaped (with its edges parallel to bishop-lines), the tables would be turned. Rooks are at their full power only when centralised, and are less powerful if nearer an edge. But bishops would be at their full power no matter where they were (except that as always they are colour-bound).

Two rooks are even more powerful than two bishops if it came down to an endgame where one opponent has two rooks they could easily use them to Roll the rooks and keep forcing the enemy king back to the edge of the board and give checkmate all by themselves without even needing the help of the king to do it. Now two Bishops and King Vs a king it takes a bit of work to figure out the tricky way to mate the enemy king and it would require the help of your king as well. It is a way trickier checkmate pattern that takes precision to execute. While the rolling rooks can be taught in just a quick session the two Bishop mate takes a substantial more sophistication and a ton more practice to become more consistent at it. The same goes for Knight and Bishop whereas a Knight and rook often kill the enemy king on their own.

Rook is a higher piece value due to the Checkmate factor . A Rook & a King can Checkmate the Opponent's King but a Bishop cannot . The Rook occupies more Squares on a Chess Board than a Bishop does . Lets take an empty board & put the Rook on a1 Square and then the Bishop. The Rook has 14 Squares under its dominance and the Bishop occupies 7 . If it is put on a Central Square the Rook has still 14 and the Bishop has max 13 .

So you see that Rook is much more worthy than the Bishop . I am not comparing any position where the relative value may be higher or lower .