George Duke wrote on Wed, Nov 5, 2008 05:06 PM UTC:
There are four basic Chess units RNBF, and three auxiliaries to perfect the
game PKQ. King, the target, is one-stepping Queen. Falcon cannot be handled
well by computer. If Falcon stands at a1 and tries for c4, the intermediate
squares are a2, b2, c3, and b3. Call them A, B, C and D respectively (D
being 'b3'). All four occupied block the move: ABCD. Any three occupied
block the move a1-c4: ABC, ABD, ACD, BCD. These two block it: AB, CD, BD.
AC does not. No one intermediate square occupied blocks it. Likewise,
certain mathematics back to Euclid cannot be handled by computer. Namely
especially, anthyphairesis. Fractions replaced anthyphairesis, which is
reciprocal subtraction, used to test incommensurability by the Greeks
2000+ years ago. For follow-up Comments here, anthyphairesis presents
patterns still regarded as apparently unpredictable. There is no pattern
as to whether sum or difference of two ratios (37:8 or -/2:1) is
commensurable or not, nor even what their anthyphairesis is. The
algorithm for 37:8 = [4,1,1,1,2] (ending therefore commensurable) is just successive divisions taking the
remainder as divisor in new ratio. Modular arithmetic.