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So we all take for granted that the value of the pieces is as follows...

P=1, N=3, B=3, R=5, Q=9

Different people over the years have attributed slightly different values to the pieces - but whilst the logic of the evaluation of the pieces makes sense, can anyone justify mathematically these values. The logic suggests that a knight can cover all squares whereas a bishop can only cover half of the squares on the board. A white squared bishop is useless against black squared pieces. A knight that is attacked may have to retreat from covering a vital square. Logically it makes sense, but not mathematically.

If you place the knight on all of the squares on the board, and add up the total number of moves it can make, you get a possible 336 moves. Averaged over 64 squares, the knight can move to 5.25 squares per move (336 / 64 = 5.25).

If you do the same with the bishop, it can move to a total of 560 squares. Averaged over 64 squares, it can move to 8.75 squares per move (560 / 64 = 8.75).

So, mathematically speaking, that would suggest that the bishop is vastly superior to the knight - more than 1 and a half times it's value.

But the bishop is limited to only half of the available squares, so that would give it an average of half the original value = 4.375.

Yet the knight is almost always considered to be the weaker of the two pieces where their value is concerned.

So, mathematically speaking (not logically), how do we justify their values of approx 3?

I haven't halved the value twice. The second picture shows the total number of possibilities for a bishop from all of the squares on the board. This is the combined values of both bishops as it covers all squares and totals 560 possible moves, an average of 8.75 moves per square. Halving this value (to account for the fact that the bishop can only access one colour) gives 4.375 squares on average.

I'm asking for a mathematical explanation, not a logical one. Yes the knight can jump, but that's already taken into account in picture 1. The total number of squares a knight can jump to, from any square on the board, is 336 - an average of 5.25 squares per move).

So, can you justify the knight and bishops value mathematically?

Some people may not understand the true value of Pieces in chess. The order from highest to lowest

is Q=9 R=5 B=3 K=3 P=1. I think that the bishop is more superior to the knight so I think the bishop is worth 4 pawns. Does anyone agree?

I'm not asking for an opinion, or a justification based on positional benefits. I'm asking for a mathematical justification of the values that we accept for the pieces.

MyRatingis1523 - you're not looking at it correctly. The total of 560 possible moves for the bishop is for all of the squares on the board, therefore BOTH bishops. The average value is also calculated on this.

Any mathematicians out there?

Sorry, you're absolutely right - I'm just very tired and not thinking straight. It's 5am here and I've been up all night. Please accept my apologies.

OP, You ask a question that can only be answered subjectively based on position and experience because how you subjectively weight the relative move properties of the pieces is important, but perhaps statistics is the branch of mathematics that will give an approximate answer if you analyze a large number of Knight vs Bishop GM games. Also just a few things wrong with your analysis about Knight superiority:

1) Bishops can cover more squares on the next turn in an open position if properly placed, what happens in the near term in chess is important.

2) Bishops can move a lot further, so they move faster in general

3) while true that Bishops only cover the same colored squares as they are, the opponents pawns will eventually have to move through that color several times before queening which is usually necessary in the endgame.

4) I assume you are just talking when each side has only one Knight or one Bishop...it is known that two Bishops are a lot better than two Knights.

This question can be answered objectively, by the means of statistics: https://www.chess.com/article/view/the-evaluation-of-material-imbalances-by-im-larry-kaufman

GM Larry Kaufman states that Bishop and Knight are equal, but Bishop's pair is worth additional 0.5 pawn, whereas Knight's value gets modified by something like 0.06 pawn * (number of pawns for the same player - 4). So in the opening position Bishop and Knight are equal, later on Knights gradually lose their value, whereas Bishops lose it abruptly when they get unpaired.

knights can jump over pieces, not just over other squares.

I think this not perfect mathematics. If you saying bishop can move on half of the squares than why did yo divide 560/64. You can divide half of square i.e. 280/32=8.75

But mathematically night is 5.25 and bishop is 8.75.

Now considering two advantages of night (1. it can cover all squares 2. It can jump over).

I just want to say it is not possible to compare these pieces mathematically.

The Reinfeld values are simply approximations. There are times where a knight might be worth much more than a bishop (e.g. a knight on an outpost on the 6th rank), and others where a bishop is worth much more than a knight (e.g. an endgame with pawns on both sides of the board). Their value is not entirely determined by the number of squares they can move to (i.e. their scope), but also in how they can move and attack on average. Determining if a knight for bishop trade is "good" is the subject of many strategic books.

In one video Garry Kasparov said that he thinks that the bishop is worth closer to 3.25

So we all take for granted that the value of the pieces is as follows...

P=1, N=3, B=3, R=5, Q=9

Different people over the years have attributed slightly different values to the pieces - but whilst the logic of the evaluation of the pieces makes sense, can anyone justify mathematically these values. The logic suggests that a knight can cover all squares whereas a bishop can only cover half of the squares on the board. A white squared bishop is useless against black squared pieces. A knight that is attacked may have to retreat from covering a vital square. Logically it makes sense, but not mathematically.

If you place the knight on all of the squares on the board, and add up the total number of moves it can make, you get a possible 336 moves. Averaged over 64 squares, the knight can move to 5.25 squares per move (336 / 64 = 5.25).

If you do the same with the bishop, it can move to a total of 560 squares. Averaged over 64 squares, it can move to 8.75 squares per move (560 / 64 = 8.75).

So, mathematically speaking, that would suggest that the bishop is vastly superior to the knight - more than 1 and a half times it's value.

But the bishop is limited to only half of the available squares, so that would give it an average of half the original value = 4.375.

Yet the knight is almost always considered to be the weaker of the two pieces where their value is concerned.

So, mathematically speaking (not logically), how do we justify their values of approx 3?

Now do the same, but put a knight/bishop in the center, and count how many moves it takes to reach each square.

The bishop's board will have a lots of N/A

The knight's will be mostly 3s.