Comments/Ratings for a Single Item
According to my theory of ranks, the values for slow pieces like the Yawn or Maw are as follows:
The Yawn has almost the same activity as the Ferz in terms of reaching enemy ranks. Thus, it reaches the 6th rank as quickly as the Ferz. But from there, the Ferz attacks squares 7+7+5+5 = 24 points, while the Yawn attacks 7+7+5 = 19 points. Thus, the Yawn should be worth 19/24 of the Ferz, or 1.2 pawns.
The Maw is an improved Wazir that lacks its slowness. While the Maw reaches its best, the 7th rank, the Ferz will only end up on the 4th. On the 4th rank, the Ferz would be worth 5+5+3+3 = 16 points, while the Maw on the 7th attacks 8+7+7+6 = 28 points. Thus, the maximum value of Maw = 28/16 of the value of the Ferz = 28/16 1.5 = 1.75 1.5 = 2.625 pawns. The practical value of Maw will be slightly lower, since from the 4th rank the Ferz can improve its position further, and Maw on the 7th has already reached its best position. However, its value should be higher than 2 pawns (even in comparison with the Ferz on the 5th rank, Maw on the 7th will show 28/20 * 1.5 = 2.1 pawns). So I think Maw is about 2.1 pawns.
Also, my theory explains why the Queen is stronger than the sum of a Rook and a Bishop, compared to a Rook, the Queen occupies more active ranks much earlier, and thus the orthogonal squares attacked by the queen matter much more than the orthogonal squares attacked by the rook.
Similarly to the Queen, the Chancellor (Rook + Knight) is active in orthogonal directions much earlier than the rook (but the Knight is able to attack the most valuable squares later than the Bishop - thats why Queen is stronger than Chancellor).
As for the Archbishop (bishop + knight), there is maximum synergy, since diagonal moves allow the relatively slow knight to be strengthened as quickly as possible. To attack the 7th rank with a knight move, the knight needs 2 moves, while the archbishop does it in 1, with a simple diagonal move.
H. G. Muller wrote on Tue, Dec 10, 2024 11:27 AM UTC in reply to Dmitry Eskin from 12:42 AM:I suppose the heuristic used by the Interactive Diagram implicitly does something similar. Most of the piece value is determined by the number of captures the piece on average can make. But this is measured on a 25% filled board, where the density of pieces of a certain color decreases linearly from their own backrank to zero just beyond the furthrest rank. The larger number of enemy pieces it will hit in the forward direction will then cause more captures for the forward moves than for similar backward moves.
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A very interesting comparison of the ferz and the wazir.
Theoretically, the ferz should be weaker due to the color weakness.
However, in practice I would draw the following conclusions:
The color weakness does not matter at all as long as the piece's range of movement is low, like the ferz. What difference does it make that the ferz cannot reach 50% of the board squares if it cannot reach 90% of the board squares due to its slowness?
Slowness is the key parameter why the ferz is much stronger than the wazir, and why the wazir on 2 ranks is much stronger than the wazir on 1.
In my model, different ranks have different values. For convenience, consider the white pieces. The first rank will be worth 1 point, the second - 2, and so on, up to 8 - eight points. These numbers show that there is a much higher chance of capturing an enemy piece on the 7th and 8th ranks than on the 1st and 2nd, because they start on the 7th and 8th ranks, and they have a long way to go to 1st and 2nd.
While the wazir from the 1st rank will go to 4th, the ferz will have time to go from 1st to 6th.
Thus, the wazir from the 4th rank will attack 1 square of the 5th rank (5 points), 2 squares of the 4th rank (4+4 = 8 points) and 1 square of the 3rd rank (3 points), a total of 5+8+3 = 16 points.
While the ferz from the 6th rank will attack 2 squares of the 7th rank (7+7 = 14 points) and 2 squares of the 5th rank (5+5 = 10 points), a total of 14+10 = 24 points.
These numbers perfectly explain why the wazir on the 1st rank is worth 1 pawn, and the ferz is worth 1.5 (24/16), that is, 1.5 times stronger.
At the same time, the wazir on the 2nd rank will be 25% more valuable (1.25 pawns), because in the example above it will have time to reach the 5th rank instead of the 4th, and will score 6 + 10 + 4 = 20 points.
I predict that the ferz on the 2nd rank will also be stronger than on the 1st, and the difference will be the same 0.25 pawns (1.75 instead of 1.5).
It also follows from this model that a double pawn move (from the starting position) has a huge significance (up to 0.25), since it allows you to attack more valuable ranks earlier, compared to a slow single move.