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?

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?