The Solitaire Army Reinspected

Abstract

Peg Solitaire soldiers-pegs for short-move on a plane square lattice. A peg P can jump over a horizontally or vertically neighboring comrade Q onto a free square, removing Q at the same time. The game starts with a configuration of pegs-a solitaire army-and the aim of the moves is usually to obtain another configuration with a prescribed property, e.g., one with a unique peg on a fixed square, or with a peg on a given remote square. For the essentials on Peg Solitaire see the definitive book Winning Ways ([2], Chapter 23), where problems of both types are treated in detail. Concerning a problem of the second kind, we have the following basic result of J. H. Conway (see [6], pp. 23-28; [2], pp. 715-717, 728; and [1]): No solitaire army stationed in the southern half-plane can send a scout into the fifth row of the northern half-plane, but an army of 20 pegs can send a scout into the fourth row. For the proof, to every square s of the plane assign a value p(s) as follows. Let obe the golden section, i.e., = (V-1)/2, so a--+ a-2 = 1. Fix a square s0 in the fifth row of the northern half-plane. For any square s, let p(s) = k, where k is the Manhattan distance between so and s (this means that so and s are exactly k horizontal or vertical one-square steps apart). Define the potential of a set of squares as the sum of values of all squares in this set, and the potential of an army as the potential of the set occupied by that army. The potential of any army with a peg standing on so is at least 1. On the other hand, the potential of the infinite army occupying all squares of the (southern) half-plane is exactly 1; we can compute it by observing that values in every column form geometric progressions with quotient . The rule of moves implies that no move can increase the potential; it follows that a finite army garrisoned in the southern half-plane cannot reach so. This kind of reasoning will occur several times in the sequel; we call it the Conway argunent. The remaining part of Conway's result can be shown simply by displaying how the army of 20 should be deployed, and how the pegs should move (see below). In fact, this means that if the front line of a solitaire army looks to the north, then it can advance four rows and no more, just four units of distance both in the Euclidean and the Manhattan sense. Armies, of course, do not always fight under such plain circumstances. Their front line may look to the southwest, for example, in which case the target may be the corner square of the first quadrant. Or, the territory to be scouted may be the half-encircled first quadrant; then the army has two perpendicular front lines, one facing north and one facing east. Or we may have two perpendicular fronts, one facing northeast and the other one northwest. FIGURE la shows an original northbound army; FIGURES lb-d show the other possibilities just mentioned. How far can the scouts be sent in cases (b), (c), and (d)? In what follows we answer these questions. The Conway argument provides upper estimates; we show that they are sharp in every case. We also prove a fact (stated in [2] without proof) concerning armies with a single mounted man. We conclude with two problems about sending scouts into an encircled ground (cf. [9]).