The theorem confirms that we have this gote traversal count and Black's 3-move traversal sequence because the conditions C ≤ A C ≤ (D + E) / 2 C ≤ A MA ≤ MC and A "G ≤ E C ≤ A": Initially, the condition A = (D + E) / 2 is the tentative gote traversal count of PA. Prove it! :) Otherwise, I ask again: how, in the general case, do we distinguish possible from impossible reversal (to start with, at least for alternating 3-move sequences)?įor proposition 10, the game tree represented by the counts is:Īn 'unsettled' position corresponds to a combinatorial game that is not a number. If other methods, such as comparing the opponent's branches or playing the difference game, are inapplicable, you have suggested MA ≤ MC as the always applicable condition for identifying or rejecting traversal of a 3-move sequence worth playing successively. Do my proposition and proof below give you enough information for this purpose? You have wanted to claim C ≤ A MA ≤ MC in the general case, but you need to prove it. However, real problem is: what is the most general case? Obviously, the requirements of PE can be relaxed to generalise the proposition with more research effort. The proposition presumes settled PD and PE and gote traversal values of PA. We want C ≤ A MA ≤ MC for the general case, but so far we cannot prove it. Some of the examples have more complicated trees with unsettled PE, an alternating gote or sente sequence for White's start, PB or PE as simple sente or long sequences resulting in series of such conditions. Furthermore, a hundred examples exhibit this relation. In particular, it relates C ≤ A MA ≤ MC, as we want. Playing the difference game yielded different results than comparing move values.Ī few weeks ago, I have proven the proposition below and its analogue for White's 3-move sequence. I think I have found a few examples rejecting this. "It is not true that if t(C) = t(A) that A reverses through C": Indeed. This is almost the opposite of the conjecture to be proven. MA // by definition of traversal move value, by (4) You have suggested that we have Black's 3-move traversal if the gote traversal move value MA is at most MC, as follows: This is expressed by the following condition:ĩ) C - V >= X - D // PA has at least the sente move value as PB White 2 should (equality: need) not reply Since PA shall have Black's 3-move traversal, PA is NOT a simple sente. This is expressed by the following conditions:ħ) B - A A > C // Black 1 gains less than White 2Ĩ) (B - V) / 2 > C // A = (B - V) / 2 expressed as simple gote count the condition means Black 1 gains less than White 2 The other case of PB being a simple gote needs to be studied later.ģ) C = (D + W) / 2 // count of simple goteĤ) MC = (D - W) / 2 // move value of simple gote Positions are annotated by P followed by a letter.Ģ) We study the case of PB being a simple sente or ambiguous. Move values are annotated by M followed by a letter. This game tree is represented by the counts of its positions.
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