Average Crowded-Board Mobility of a Cannon Piece: Numbers
Spreadsheet-style Calculation
MOVE ON-BOARD LEGAL PRODUCT SUBTOTAL Pawns
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(0,2) 3.0 0.3 0.9 0.9 0.5
(0,3) 2.5 0.42 1.05 1.95 1.1142857142
(0,4) 2.0 0.441 0.882 2.832 1.6182857142
(0,5) 1.5 0.4116 0.6174 3.4494 1.9710857142
(0,6) 1.0 0.36015 0.36015 3.80955 2.1768857142
(0,7) 0.5 0.302526 0.151263 3.960813 2.2633217142
(2,2) 2.25 0.3 0.675 0.675 0.3857142857
(3,3) 1.5625 0.42 0.65625 1.33125 0.7607142857
(4,4) 1.0 0.441 0.441 1.77225 1.0127142857
(5,5) 0.5625 0.4116 0.231525 2.003775 1.1450142857
(6,6) 0.25 0.36015 0.0900375 2.0938125 1.1964642857
(7,7) 0.0625 0.302526 0.018907875 2.112720375 1.2072687857
pQ2 1.575 0.8857142857
pQ3 3.28125 1.875
pQ4 4.60425 2.631
pQ5 5.453175 3.1161
pQ6 5.9033625 3.37335
pQ7 6.073533375 3.4705905
Rook calculations for different values of "Empty"
Probable cannon-
Empty Rook Rook RpR
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0.500000 6.007812 4.078125 10.085938
0.515625 6.137599 4.097157 10.234757
0.531250 6.272215 4.113005 10.385220
0.546875 6.411872 4.125298 10.537170
0.562500 6.556794 4.133637 10.690431
0.578125 6.707211 4.137598 10.844809
0.593750 6.863364 4.136724 11.000088
0.609375 7.025506 4.130527 11.156033
0.625000 7.193895 4.118488 11.312384
0.640625 7.368805 4.100054 11.468859
0.656250 7.550518 4.074633 11.625151
0.671875 7.739326 4.041601 11.780927
0.687500 7.935536 4.000292 11.935828
0.703125 8.139463 3.950001 12.089464
0.718750 8.351437 3.889981 12.241418
0.734375 8.571799 3.819442 12.391241
0.750000 8.800903 3.737549 12.538452
0.765625 9.039117 3.643420 12.682537
0.781250 9.286822 3.536125 12.822947
0.796875 9.544412 3.414684 12.959096
0.812500 9.812297 3.278064 13.090361
0.828125 10.090900 3.125180 13.216080
0.843750 10.380660 2.954891 13.335551
0.859375 10.682032 2.765996 13.448028
0.875000 10.995485 2.557240 13.552725
0.890625 11.321507 2.327300 13.648807
0.906250 11.660599 2.074796 13.735395
0.921875 12.013282 1.798280 13.811561
0.937500 12.380093 1.496235 13.876327
Queen calculations for different values of "Empty"
Probable cannon-
Empty Queen Queen QpQ
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0.500000 10.754883 6.544922 17.299805
0.515625 10.962813 6.552653 17.515466
0.531250 11.177727 6.555013 17.732741
0.546875 11.399910 6.551487 17.951396
0.562500 11.629654 6.541524 18.171178
0.578125 11.867265 6.524541 18.391806
0.593750 12.113061 6.499914 18.612974
0.609375 12.367370 6.466982 18.834352
0.625000 12.630535 6.425044 19.055579
0.640625 12.902911 6.373354 19.276265
0.656250 13.184865 6.311125 19.495990
0.671875 13.476779 6.237523 19.714302
0.687500 13.779049 6.151666 19.930715
0.703125 14.092085 6.052624 20.144708
0.718750 14.416312 5.939413 20.355724
0.734375 14.752169 5.810998 20.563167
0.750000 15.100113 5.666290 20.766403
0.765625 15.460615 5.504141 20.964756
0.781250 15.834164 5.323344 21.157508
0.796875 16.221264 5.122632 21.343896
0.812500 16.622438 4.900674 21.523113
0.828125 17.038226 4.656075 21.694301
0.843750 17.469186 4.387371 21.856557
0.859375 17.915896 4.093029 22.008925
0.875000 18.378950 3.771445 22.150395
0.890625 18.858966 3.420940 22.279905
0.906250 19.356577 3.039759 22.396336
0.921875 19.872441 2.626069 22.498509
0.937500 20.407234 2.177955 22.585188
Bishop calculations for different values of "Empty"
Probable cannon-
Empty Bishop Bishop BpB
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0.500000 4.747070 2.466797 7.213867
0.515625 4.825213 2.455496 7.280709
0.531250 4.905512 2.442008 7.347521
0.546875 4.988037 2.426189 7.414226
0.562500 5.072860 2.407887 7.480747
0.578125 5.160054 2.386943 7.546997
0.593750 5.249696 2.363190 7.612886
0.609375 5.341865 2.336455 7.678320
0.625000 5.436640 2.306555 7.743195
0.640625 5.534106 2.273301 7.807406
0.656250 5.634347 2.236492 7.870839
0.671875 5.737453 2.195922 7.933375
0.687500 5.843513 2.151374 7.994888
0.703125 5.952622 2.102623 8.055245
0.718750 6.064875 2.049432 8.114306
0.734375 6.180370 1.991556 8.171926
0.750000 6.299210 1.928741 8.227951
0.765625 6.421498 1.860721 8.282219
0.781250 6.547342 1.787219 8.334561
0.796875 6.676852 1.707948 8.384801
0.812500 6.810142 1.622610 8.432752
0.828125 6.947326 1.530895 8.478221
0.843750 7.088526 1.432480 8.521007
0.859375 7.233864 1.327033 8.560897
0.875000 7.383465 1.214205 8.597670
0.890625 7.537459 1.093639 8.631098
0.906250 7.695978 0.964962 8.660941
0.921875 7.859159 0.827789 8.686948
0.937500 8.027141 0.681720 8.708861
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