MAZE was introduced in [Albert et al. 2007], but apart from a few scattered observations, nothing was known about the values of the game. In the original article, MAZE is played on a rectangular grid oriented 45◦ to the horizontal.

The token starts at the top of the board and highlighted edges are walls that may not be crossed. Left is allowed to move a token any number of cells in a southwesterly direction and Right is allowed to move similarly in a southeasterly direction. However, for ease of referring to specific places in the position, we re-orient the sides parallel to the page so that Left moves downward and Right moves to the right; see Figure 1. One interesting feature is that any number of consecutive Left (Right) moves also can be accomplished in one move. This feature had been noted in several games, including HACKENBUSH strings [Berlekamp et al. 2001], and given the name of option-closed in [Nowakowski and Ottaway 2011], a reference we henceforth abbreviate as [NO]. Siegel [2011] notes that the partial order of option-closed games born on day n forms a planar lattice.

In two-player combinatorial games, the last player to move either wins (normal play) or loses (misère play). Traditionally, normal play games have garnered more attention due to the group structure which arises on such games. Less work has been done with games played under the misère play convention, Just as in normal play, misère games can be placed in equivalence classes, where two games G and H are equivalent if the outcome class of G + K is the same as the outcome class of H+K for all games K. However, Conway showed that, unlike in normal play, these misère equivalence classes are sparsely populated, making the analysis of misère games under such equivalence classes far less useful than their normal play counterparts [ONAG]. Even though these equivalence classes are sparse, Conway developed a method, called genus theory, for analyzing impartial games played under the misère play convention [Allen 2006; WW; ONAG]. For years, this was the only universal tool available for those studying misère games.

In [Plambeck 2009; 2005; Plambeck and Siegel 2008; Siegel 2006; 2015b], many results regarding impartial misère games have been achieved. These results were obtained by taking a game, restricting the universe in which that game was played, and obtaining its misère quotient. However, while, as Siegel [2015a] says “a partizan generalization exists”, few results have been obtained regardingthe structure of the misère quotients which arise from partizan games.

For a game G ={GL|GR}, we define Ḡ = {GR|GL}. Those familiar with normal play will notice that under the normal play convention rather than Ḡ, we would generally write̶G. In normal play, this nomenclature is quite sensible as G+(-G)=0 [Albert et al. 2007], giving us the Tweedledum–Tweedledee principle; the second player can always win the game G+(-G) by mimicking the move of the first player, but in the other component. However, in misère play, not only does the Tweedledum–Tweedledee strategy often fail, G + Ḡ is not necessarily equivalent to 0. For example, *2+*2=*2+*2 is not equivalent to 0 [Allen 2006; WW]. However, having the property that G+Ḡ is equivalent to 0 is much desired, as it gives a link to which partizan misère games may behave like their normal counterparts.

The Rat game is played on 3 piles of tokens by 2 players who play alternately. Positions in the game are denoted throughout in the form (x, y, z), with 0 ≤ x ≤ y ≤ z, and moves in the form (x, y, z)→ (u, v, w), where of course also 0 ≤ u ≤ v ≤ w (see below). The player first unable to move—because the position is (0, 0, 0)—loses; the opponent wins.

Disjunctive compounds of short combinatorial games have been studied for many years under a variety of assumptions. A structure theory for normal-play impartial games was established in the 1930s by the Sprague–Grundy theorem [Grundy 1939; Sprague 1935; 1937]. Every such game G is equivalent to a Nim-heap, and the size of this heap, known as the nim value of G, completely describes the behavior of G in disjunctive sums. The Sprague–Grundy theorem underpins virtually all subsequent work on impartial combinatorial games.

Decades later, Conway generalized the Sprague–Grundy theorem in two directions [Berlekamp et al. 2003; Conway 2001]. First, he showed that every partizan game G can be assigned a value that exactly captures its disjunctive behavior, and this value is represented by a unique simplest form for G. Conway’s game values are partizan analogues of nim values, and his simplest form theorem directly generalizes the Sprague–Grundy theorem.

Conway also introduced a misère-play analogue of the Sprague–Grundy theorem. He showed that every impartial game G is represented by a unique misère simplest form [Conway 2001]. Unfortunately, in misère play such simplifications tend to be weak, and as a result the canonical theory of misère games is less useful in practice than its normal-play counterparts.

In each case—normal-play impartial, normal-play partizan, and misère-play impartial—the identification of simplest forms proved to be a key result, at once establishing a structure theory and opening the door to further investigations. In this paper, we prove an analogous simplest form theorem for the misère-play partizan case. The proof integrates techniques drawn from each of Conway’s advances, together with a crucial lemma from [Mesdal and Ottaway 2007].

Hex is well-known for the simplicity of its rules and the complexity of its play. Nash’s strategy-stealing argument shows that a winning strategy for the first player exists, but finding such a strategy is intractable by current methods on large boards. It is not known, for instance, whether the center hex is a winning first move on an odd size board. The development of artificial intelligence for Hex is a notoriously rich and challenging problem, and has been an active area of research for over thirty years [Davis 1975/76; Nishizawa 1976; Anshelevich 2000; 2002a; 2002b; Cazenave 2001; Rasmussen and Maire 2004], yet the best programs play only at the level of an intermediate human [Melis and Hayward 2003]. Complete analysis of Hex is essentially intractable; the problem of determining which player has a winning strategy from a given board position is PSPACE-complete [Reisch 1981], and the problem of determining whether a given empty hex is dead, or irrelevant to the outcome of the game, is NP-complete [Björnsson et al. 2007]. Some recent research has focused on explicit solutions for small boards [Hayward et al. 2004; Hayward et al. 2005], but it is unclear whether such techniques will eventually extend to the standard 11x11 board.

Hex was invented independently by Piet Hein [1942] and John Nash [1952]. The game is played by two players, Black and White, on a board with hexagonal cells. The players alternate turns, colouring any single uncoloured cell with their colour. The winner is the player who creates a path of her colour connecting her two opposing board sides. See Figure 1.

Hein and Nash observed that Hex cannot end in a draw [Hein 1942; Nash 1952]: exactly one player has a winning path if all cells are coloured [Beck et al. 1969]. Also, an extra coloured cell is never disadvantageous for the player with that colour [Nash 1952]. For n x n boards, Nash showed the existence of a first-player winning strategy [1952]; however, his proof reveals nothing about the nature of such a strategy. For 8x8 and smaller boards, computer search can find all winning first moves [Hayward et al. 2004; Henderson et al. 2009]. For the 9x9 board, Yang found by human search that moving to the centre cell is a winning first move.

This paper is about the complexity of combinatorial games. Its main contribution is showing constructively that a large class of games whose complexity was hitherto unknown and its best winning strategy was exponential, is actually solvable in polynomial time.

Sprouts is a two-player pencil-and-paper game invented in 1967 in the University of Cambridge by John Conway and Michael Paterson [Gardner 1967]. The game starts with p spots and players alternately connect the spots by drawing curves between them, adding a new spot on each curve drawn. A new curve cannot cross or touch any existing one, leading necessarily to a planar graph. The first player who cannot play loses.