The Focardi and Luccio Analysis of 7+

The Focardi and Luccio paper entitled "A New Analysis Technique for the Sprouts Game" has two parts. The first part develops a formula for the maximum number of survivors any given sphere can yield. I have not studied this part of the paper.

The second part purports to give a logically exhaustive analysis of 7+ using the survivor motif and the formula developed in the first part of the paper. (The formula is used to certify that particular spheres could not possibly produce more than one survivor.) The analysis turns out to be pretty good but not quite logically exhaustive.

The authors proceed by analyzing several diagrams. Each of the diagrams consists of a circle with one or more spots on the circle and with one or more spots inside the circle. In each case the authors determine that "player B", the person playing the even numbered moves in the given diagram, can force the production of some particular number of survivors. For several of these diagrams the authors also point out that none of the survivors produced are spots on the circle and also that in the given analysis player B has not "made use of" any spot on the circle. These latter observations are evidently intended to help with situations in the 7+ analysis in which two diagrams are combined and share a pivot, but this assistance proves inadequate, as detailed below.

Armed with the results of the analyses of the several diagrams, the authors attempt to prove that in the 7+ game Right can produce five survivors. The most obvious hole in their demonstration shows up in the line 7+ 1(8)2 1(9)1[2,3]. Here the authors claim that play in the spheres S(2) and S(4) can in each case follow the analyses they have supplied for the related diagrams. Unfortunately, though, the move 4(10)9 affects both spheres simultaneously such that Right would be required to make two moves at once in order to stay in the authors' "book". Right can win this position, but the authors do not show how.

The authors also fail to provide needed guarantees that in positions combining two or more of the previously analyzed diagrams Right will in fact be "player B" in every diagram. They have not addressed the question of whether play in one diagram might invert the move order in another diagram. This omission keeps the authors' argument from being logically tight, even apart from the problem mentioned above.

Despite these lapses, the second part of the paper does a fairly good job of showing how to play 7+. The authors' goal of proving their result with mathematical rigor is not attained, but a reasonably comprehensive repertory of sound lines is presented.

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