Average Crowded-Board Mobility of a Grasshopper Piece.

The Grasshopper power is quite weak, so it might make an interesting addition to some other piece. In order for a Grasshopper to move or capture, it must jump over one other piece, and land on the square just after that piece.

I'll use good old 0.7 as the "normal" probability that a square will be empty.

So, let's do the calculation by hand. For each of the possible grasshopper-Queen moves, write down the probability that the square is on the board (this is the "Average Empty-Board Mobility", given as "((w-x)*(h-y)/(w*h))*N"), then write down the probability that the square will be legal by grasshopper rules (in order to go to (0, n), the grassopper must find the square at n-minus-one occupied, and all the squares from 1 to n-minus-two must be empty), then write down the product of those two, then do the appropriate sums. Finally, divide each sum by 1.75 (which is 5.25/3) to get the value in Pawns, more or less.

After calculating the values to ten decimal points, we'll embark on the process of rounding the results and using common sense to see how reasonable the numbers seemed to be.

This step is less precise than the calculations, but is more important in getting good results.

We won't bother to calculate the value of the grasshopper Knight-rider! (But you'll know how to do it if you want to.)

=====    ========  =======     ========     ========     ======
(0,2)    3.0       0.3         0.9          0.9          0.5
(0,3)    2.5       0.21        0.525        1.425        0.8142857142
(0,4)    2.0       0.147       0.294        1.719        0.9822857142
(0,5)    1.5       0.1029      0.15435      1.87335      1.0704857142
(0,6)    1.0       0.07203     0.07203      1.94538      1.1116457142
(0,7)    0.5       0.050421    0.0252105    1.9705905    1.1260517142

(2,2)    2.25      0.3         0.675        0.675        0.3857142857
(3,3)    1.5625    0.21        0.328125     1.003125     0.5732142857
(4,4)    1.0       0.147       0.147        1.150125     0.6572142857
(5,5)    0.5625    0.1029      0.05788125   1.20800625   0.6902892857
(6,6)    0.25      0.07203     0.0180075    1.22601375   0.7005792857
(7,7)    0.0625    0.050421    0.0031513125 1.2291650625 0.7023800357

gQ2                                         1.575        0.8857142857
gQ3                                         2.428125     1.3874999999
gQ4                                         2.869125     1.6394999999
gQ5                                         3.08135625   1.760775
gQ6                                         3.17139375   1.812225
gQ7                                         3.1997555625 1.82843175

So, the calculations say that a grasshopper Rook is worth 1.1 Pawns, a grasshopper Bishop is worth 0.7 Pawns, and a grasshopper Queen is worth 1.8; or in Point Count Chess terms, a grasshopper Rook is 3 points, a grasshopper Bishop is two, and a grasshopper Queen is five.

That really cuts through the decimal points, doesn't it? Remember that the Point Count Chess "point" is more or less the Quantum of Advantage, so it makes sense to round off the results this grossly.

Let's also remember that the NA, NF, ND, and NW are equal to each other in strength although the absolute mobilities of the W, A, F, and D are very different (3.5, 2.25, 3.06, and 3.0). The Grasshopper Queen's mobility of 3.2 is in this range, but her advantage of having long-distance moves in 8 directions probably makes her a more powerful piece in her own right, and a more powerful addition to other pieces, than any of them. At least, in the opening and midgame. In the endgame, the Grasshopper Queen's value becomes zero, or at least very nearly zero.

The gQ is expected to be worth more than the separate gR and gB; so we could guess a value of two Pawns.

In the opening, a grasshopper piece's penetrating power means that it might be used to win material at the start of a game. Thus, if we have a piece FgQ (Ferz plus grasshopper Queen) on b1, we can start the game with the Grasshopper move 1. (FgQ)b1-d3 [attacks d8], d7-d6?? [d7-d5 was forced] 2. (FgQ)d3-e4+ forking a8.

Remember that these numbers are not "official" estimates of the values of these pieces. The numbers are always wrong.

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