Nim

Nim

Deconstructed by: Valentina Tamer

Nim is a mathematical strategy game of perfect information. It is of uncertain origin, but documents from 16th century Europe show first references. Instead of playing it by removing matches, one can also cross out lines on paper, making this a pen-and-paper game if desired. All rules stated here remain he same in this case.

Goal: Usually played as a misère game, the goal is to not take the last match remaining. (One variation is the opposite goal, played as 'normal play': The one to take the last object wins)

Core Mechanic: The players remove matches off a table in turns, in a manner of an impartial game [=a game in which the allowable moves depend on the position of the objects and not the player who is currently moving].

Action: The player takes any number of matches off the field, whereas in one turn, only one row can be taken from.

Space of the game: 16 0-dimensional discrete cells, connected in one big 0-dimensional grid [in order to compare whether a match is the last remaining]. Within this grid, there further exclusive connections of 0-dimensional rows in the following pairs: 1, 3, 5 and 7 [see attachment].

Objects: 16 Matches

Attributes: "Last Match" (If false, the one who takes it immediately loses. If true, the player may take it without further consequence except for its state changing to 'off the field')

State: On the field (starter state) & Off the field (end state, cannot be changed unless a new game is started)

Rules: A turn-based impartial 2-player game. [Operational/Written:] Every player can take as many matches from one row as he wants, but at least one and only out of one row in each turn. ([Foundational:] Take x matches from one row, whereas 0<x≤y (y= number of remaining matches in one row))

Skills Players learn: The game is a challenge to the (real) mental skills of the players. In order to consciously contribute to winning a game, the player's mathematical thinking and analytical attentiveness are needed. Without trying to find a logic, an algorithm, behind the events, it will be decided by instincts and chance, but human need to bring order to chaos will usually result in analytical activity.

The basis behind the game is the Sprague-Grundy theorem: "Every impartial game is equivalent to a nim heap of a certain size that yields to the same outcome when played parallel with other normal play in impartial games" In other words: The matches can be divided into groups of 4, 2 and 1, whereas an even number of each group means an advantage for the player who was responsible for bringing the field into that state. In the beginning, the heaps are even:

row 1 (1): 1

row 2 (3): 2+1

row 3 (5): 4+1

row 4 (7): 4+2+1

Therefore: 4,4/ 2,2/ 1,1/ 1,1. No heap is left over without a partner.

As soon as one heap stands on its own, the enemy potentially has an advantage. When the player aims to always set the field to an even numbers of groups of 4, 2 and 1, he directs towards winning the game.

An example of an uneven field would be this:

row 1 (1): 1

row 2 (3): 2+1

row 3 (5): 4+1

row 4 (6): 4+2

Therefore: 4,4/ 2,2/ 1,1/ and 1 is left without a partner. In order to fix this, that 1 would have to be taken away.

Variations:

The subtraction Game: The maximal number of objects allowed to be removed from the game is restricted, depending on what number the players decide on.

Multiple-heap rule: Here, besides moving any number from one single heap, the player can also remove the same number of objects from each heap.

Circular Nim: Any number of objects is placed in a circle and the players alternatively remove 1,2 or 3 adjacent objects. The one to take the last one loses.

The 21 Game: In this game, the players take turns in saying a number, starting with one. With each turn, the next player increases the number by 1,2 or 3, but must not exceed 21. The one to say 21 loses.