In the mathematical theory of games, for an impartial game under the misère convention, finding the indistinguishability quotient of all the positions in a game opens the theoretical possibility of predicting the outcome of the sum of any number of games (positions) in a game.
The indistinguishability quotient of a game is expressed as the product of a number of symbols. Usually, the quotient of games is found bottom up, with quotients for subsequently larger positions found.
When a position cannot be expressed as the sum of other smaller positions, a new symbol is created to represent it. Other positions which can be expressed as a sum of other positions are given a quotient that is the product of all the positions that can be used to express it. A position can be expressed as a sum of other positions if it is equivalent (see Sprague–Grundy theorem).
A set of equations with equivalent quotients is also kept to keep the quotients shorter. For example, with many misère games, 2 + 2 + 2 = 2 is true. Then, if 2 is assigned a quotient of a, the equation can be expressed as a3 = a.
Finally, a kernel of quotients which are losing positions is kept.
With this information, the outcome of any sum of games can be found by multiplying the quotient of each individual game together, simplifying with the equations found, and looking up whether the quotient is in the kernel.