Nim

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For other uses, see Nim (disambiguation).

Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap.

Variants of Nim have been played since ancient times.^{[citation needed]} The game is said to have originated in China (it closely resembles the Chinese game of "Jianshizi", or "picking stones"), but the origin is uncertain; the earliest European references to Nim are from the beginning of the 16th century. Its current name was coined by Charles L. Bouton of Harvard University, who also developed the complete theory of the game in 1901, but the origins of the name were never fully explained. The name is probably derived from German nimm meaning "take [imperative]", or the obsolete English verb nim of the same meaning.

Nim can be played as a misère game, in which the player to take the last object loses. Nim can also be played as a normal play game, which means that the person who makes the last move (i.e., who takes the last object) wins. This is called normal play because most games follow this convention, even though Nim usually does not.

Normal play Nim (or more precisely the system of nimbers) is fundamental to the Sprague–Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a Nim heap that yields the same outcome when played in parallel with other normal play impartial games (see disjunctive sum).

While all normal play impartial games can be assigned a nim value, that is not the case under the misère convention. Only tame games can be played using the same strategy as misère nim.

A version of Nim is played—and has symbolic importance—in the French New Wave film Last Year at Marienbad (1961).

At the 1940 New York World's Fair Westinghouse displayed a machine, the Nimatron, that played Nim.^[1] It was also one of the first ever electronic computerized games (1952). Herbert Koppel, Eugene Grant and Howard Bailer, engineers from the W.L. Maxon Corporation, developed a machine weighing 50 pounds which played Nim against a human opponent and regularly won.^[2]

Nim is a special case of a poset game where the poset consists of disjoint chains (the heaps).

Game play and illustration

The normal game is between two players and played with three heaps of any number of objects. The two players alternate taking any number of objects from any single one of the heaps. The goal is to be the last to take an object. In misère play, the goal is instead to ensure that the opponent is forced to take the last remaining object.

The following example game is played between fictional players Bob and Alice who start with heaps of three, four and five objects. The winning strategy is for a player to always leave an even total number of 1's, 2's, and 4's. In the example, Bob implements this strategy.

Sizes of heaps  Moves
A B C
 
3 4 5           Bob   takes 2 from A
1 4 5           Alice takes 3 from C
1 4 2           Bob   takes 1 from B
1 3 2           Alice takes 1 from B
1 2 2           Bob   takes entire A heap, leaving two 2s.
0 2 2           Alice takes 1 from B
0 1 2           Bob   takes 1 from C leaving two 1s. (In misère play he would take 2 from C leaving (0, 1, 0).)
0 1 1           Alice takes 1 from B
0 0 1           Bob   takes entire C heap and wins.

Mathematical theory

Nim has been mathematically solved for any number of initial heaps and objects, and there is an easily calculated way to determine which player will win and what winning moves are open to that player. In a game that starts with heaps of three, four, and five, the first player will win with optimal play, whether the misère or normal play convention is followed.

The key to the theory of the game is the binary digital sum of the heap sizes, that is, the sum (in binary) neglecting all carries from one digit to another. This operation is also known as "exclusive or" (xor) or "vector addition over GF(2)". Within combinatorial game theory it is usually called the nim-sum, as will be done here. The nim-sum of x and y is written x ⊕ y to distinguish it from the ordinary sum, x + y. An example of the calculation with heaps of size 3, 4, and 5 is as follows:

Binary  Decimal
 
  011₂    3₁₀    Heap A
  100₂    4₁₀    Heap B
  101₂    5₁₀    Heap C
  ---
  010₂    2₁₀    The nim-sum of heaps A, B, and C, 3 ⊕ 4 ⊕ 5 = 2

An equivalent procedure, which is often easier to perform mentally, is to express the heap sizes as sums of distinct powers of 2, cancel pairs of equal powers, and then add what's left:

3 = 0 + 2 + 1 =     2   1      Heap A
4 = 4 + 0 + 0 = 4              Heap B
5 = 4 + 0 + 1 = 4       1      Heap C
---
2 =                 2          What's left after canceling 1s and 4s

In normal play, the winning strategy is to finish every move with a Nim-sum of 0. This is always possible if the Nim-sum is not zero before the move. If the Nim-sum is zero, then the next player will lose if the other player does not make a mistake. To find out which move to make, let X be the Nim-sum of all the heap sizes. Take the Nim-sum of each of the heap sizes with X, and find a heap whose size decreases. The winning strategy is to play in such a heap, reducing that heap to the Nim-sum of its original size with X. In the example above, taking the Nim-sum of the sizes is X = 3 ⊕ 4 ⊕ 5 = 2. The Nim-sums of the heap sizes A=3, B=4, and C=5 with X=2 are

A ⊕ X = 3 ⊕ 2 = 1 [Since (011) ⊕ (010) = 001 ]

B ⊕ X = 4 ⊕ 2 = 6

C ⊕ X = 5 ⊕ 2 = 7

The only heap that is reduced is heap A, so the winning move is to reduce the size of heap A to 1 (by removing two objects).

As a particular simple case, if there are only two heaps left, the strategy is to reduce the number of objects in the bigger heap to make the heaps equal. After that, no matter what move your opponent makes, you can make the same move on the other heap, guaranteeing that you take the last object.

When played as a misère game, Nim strategy is different only when the normal play move would leave no heap of size two or larger. In that case, the correct move is to leave an odd number of heaps of size one (in normal play, the correct move would be to leave an even number of such heaps).

In a misère game with heaps of sizes three, four and five, the strategy would be applied like this:

A B C Nim-sum
 
3 4 5 010₂=2₁₀   I take 2 from A, leaving a sum of 000, so I will win.
1 4 5 000₂=0₁₀   You take 2 from C
1 4 3 110₂=6₁₀   I take 2 from B
1 2 3 000₂=0₁₀   You take 1 from C
1 2 2 001₂=1₁₀   I take 1 from A
0 2 2 000₂=0₁₀   You take 1 from C
0 2 1 011₂=3₁₀   The normal play strategy would be to take 1 from B, leaving an even number (2)
                 heaps of size 1.  For misère play, I take the entire B heap, to leave an odd
                 number (1) of heaps of size 1.
0 0 1 001₂=1₁₀   You take 1 from C, and lose.

The previous strategy for a misère game can be easily implemented (for example in Python, below).

def nim(heaps, misere=True):
	"""Computes next move for Nim in a normal or misère (default) game, returns tuple (chosen_heap, nb_remove)"""
	X = reduce(lambda x,y: x^y, heaps)
	if X == 0: # Will lose unless all non-empty heaps have size one
		if max(heaps) > 1:
			print "You will lose :("
		for i, heap in enumerate(heaps):
			if heap > 0: # Empty any (non-empty) heap
				chosen_heap, nb_remove = i, heap
				break
	else:
		sums = [t^X < t for t in heaps]
		chosen_heap = sums.index(True)
		nb_remove = heaps[chosen_heap] - (heaps[chosen_heap]^X)
		heaps_twomore = 0
		for i, heap in enumerate(heaps):
			n = heap-nb_remove if chosen_heap == i else heap
			if n>1: heaps_twomore += 1
		# If move leaves no heap of size 2 or larger, leave an odd (misère) or even (normal) number of heaps of size 1
		if heaps_twomore == 0:
			chosen_heap = heaps.index(max(heaps))
			heaps_one = sum(t==1 for t in heaps)
			# misère (resp. normal) strategy: if it is even (resp. odd) make it odd (resp. even), else do not change
			nb_remove = heaps[chosen_heap]-1 if heaps_one%2!=misere else heaps[chosen_heap]
	return chosen_heap, nb_remove

Proof of the winning formula

The soundness of the optimal strategy described above was demonstrated by C. Bouton.

Theorem. In a normal Nim game, the player making the first move has a winning strategy if and only if the nim-sum of the sizes of the heaps is nonzero. Otherwise, the second player has a winning strategy.

Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+), and also satisfies an additional property, x ⊕ x = 0 (technically speaking, the nonnegative integers under ⊕ form an Abelian group of exponent 2).

Let x₁, ..., x_n be the sizes of the heaps before a move, and y₁, ..., y_n the corresponding sizes after a move. Let s = x₁ ⊕ ... ⊕ x_n and t = y₁ ⊕ ... ⊕ y_n. If the move was in heap k, we have x_i = y_i for all i ≠ k, and x_k > y_k. By the properties of ⊕ mentioned above, we have

    t = 0 ⊕ t
      = s ⊕ s ⊕ t
      = s ⊕ (x₁ ⊕ ... ⊕ x_n) ⊕ (y₁ ⊕ ... ⊕ y_n)
      = s ⊕ (x₁ ⊕ y₁) ⊕ ... ⊕ (x_n ⊕ y_n)
      = s ⊕ 0 ⊕ ... ⊕ 0 ⊕ (x_k ⊕ y_k) ⊕ 0 ⊕ ... ⊕ 0
      = s ⊕ x_k ⊕ y_k
 
(*) t = s ⊕ x_k ⊕ y_k.

The theorem follows by induction on the length of the game from these two lemmas.

Lemma 1. If s = 0, then t ≠ 0 no matter what move is made.

Proof: If there is no possible move, then the lemma is vacuously true (and the first player loses the normal play game by definition). Otherwise, any move in heap k will produce t = x_k ⊕ y_k from (*). This number is nonzero, since x_k ≠ y_k.

Lemma 2. If s ≠ 0, it is possible to make a move so that t = 0.

Proof: Let d be the position of the leftmost (most significant) nonzero bit in the binary representation of s, and choose k such that the dth bit of x_k is also nonzero. (Such a k must exist, since otherwise the dth bit of s would be 0.) Then letting y_k = s ⊕ x_k, we claim that y_k < x_k: all bits to the left of d are the same in x_k and y_k, bit d decreases from 1 to 0 (decreasing the value by 2^d), and any change in the remaining bits will amount to at most 2^d−1. The first player can thus make a move by taking x_k − y_k objects from heap k, then

t = s ⊕ x_k ⊕ y_k           (by (*))
  = s ⊕ x_k ⊕ (s ⊕ x_k)
  = 0.

The modification for misère play is demonstrated by noting that the modification first arises in a position that has only one heap of size 2 or more. Notice that in such a position s ≠ 0, therefore this situation has to arise when it is the turn of the player following the winning strategy. The normal play strategy is for the player to reduce this to size 0 or 1, leaving an even number of heaps with size 1, and the misère strategy is to do the opposite. From that point on, all moves are forced.

Other variations of Nim

The subtraction game S(1,2,...,k)

In another game which is commonly known as Nim (but is better called the subtraction game S (1,2,...,k)), an upper bound is imposed on the number of objects that can be removed in a turn. Instead of removing arbitrarily many objects, a player can only remove 1 or 2 or ... or k at a time. This game is commonly played in practice with only one heap (for instance with k = 3 in the game Thai 21 on Survivor: Thailand, where it appeared as an Immunity Challenge).

Bouton's analysis carries over easily to the general multiple-heap version of this game. The only difference is that as a first step, before computing the Nim-sums, we must reduce the sizes of the heaps modulo k + 1. If this makes all the heaps of size zero (in misère play), the winning move is to take k objects from one of the heaps. In particular, in ideal play from a single heap of n objects, the second player can win if and only if

n ≡ 0 (mod k+1) (in normal play), or

n ≡ 1 (mod k+1) (in misère play).

This follows from calculating the nim-sequence of S(1,2,...,k),

$0.123\ldots k0123\ldots k0123\dots ={\dot 0}.123\ldots {\dot {k}},\,$

from which the strategy above follows by the Sprague–Grundy theorem.

The 21 game

The game "21" is played as a misère game with any number of players who take turns saying a number. The first player says "1" and each player in turn increases the number by 1, 2, or 3, but may not exceed 21; the player forced to say "21" loses. This can be modeled as a subtraction game with a heap of 21–n objects.
The winning strategy for this game is to say a multiple of 4 and after that it is guaranteed that the other player will have to say 21, barring a mistake from the first player.
This game has a Sprague–Grundy value of zero, i.e., it is biased in favor of the second player as s/he can get to 4 first and then control the game from there, as no matter what, the first player will never be able to say a multiple of 4 as s/he is only allowed increments of either 1, 2 or 3.
The 21 game can also be played with different numbers, like "Add up to 5 and Lose on 34".
Proof (via a sample game of 21)-

Player Number

 1           1
 2           4
 1        5,6 or 7
 2           8
 1        9,10 or 11
 2          12
 1        13,14 or 15
 2          16
 1        17,18 or 19
 2          20
 1          21

The 100 game

A similar version is the "100 game": two players start from 0 and alternatively add a number from 1 to 10 to the sum. The player who reaches 100 wins. The winning strategy is to reach a number in which the digits are subsequent (e.g. 01, 12, 23, 34,...) and control the game by jumping through all the numbers of this sequence. Once reached 89, the opponent has lost (he can only tell numbers from 90 to 99, and the next answer can in any case be 100).

A multiple-heap rule

Circular Nim

Grundy's game

In Grundy's game, another variation of Nim, a number of objects are placed in an initial heap, and two players alternately divide a heap into two nonempty heaps of different sizes. Thus, six objects may be divided into piles of 5+1 or 4+2, but not 3+3. Grundy's game can be played as either misère or normal play.

Greedy Nim

See Greedy Nim.

References

↑ Flesch, Rudolf (1951). The Art of Clear Thinking. New York: Harper and Brothers Publishers. p. 3.
↑ Grant, Eugene F.; Lardner, Rex (August 2, 1952). "The Talk of the Town – It". The New Yorker.

W. W. Rouse Ball: Mathematical Recreations and Essays, The Macmillan Company, 1947.
John D. Beasley: The Mathematics of Games, Oxford University Press, 1989.
Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy: Winning Ways for your Mathematical Plays, Academic Press, Inc., 1982.
C. L. Bouton: Nim, a game with a complete mathematical theory, Annals of Mathematics 3 (1901–02), 35–39.
Manfred Eigen and Ruthild Winkler: Laws of the Game, Princeton University Press, 1981.
Walter R. Fuchs: Computers: Information Theory and Cybernetics, Rupert Hart-Davis Educational Publications, 1971.
G. H. Hardy and E. M. Wright: An Introduction to the Theory of Numbers, Oxford University Press, 1979.
Edward Kasner and James Newman: Mathematics and the Imagination, Simon and Schuster, 1940.
M. Kaitchik: Mathematical Recreations, W. W. Norton, 1942.
Donal D. Spencer: Game Playing with Computers, Hayden Book Company, Inc., 1968.

External links

1952: 50-pound computer plays Nim- The New Yorker magazine "Talk of the Town" August, 1952.:
1958: Popular Electronics magazine Jan 1958, article "Win at NIM with Debicon" – an electrical circuit that shows lights to assist in winning a game.
Nim Flash on Kongregate
Nim in JavaScript – including its historical aspect at Archimedes-lab.org.
Nim in JavaScript IE7 and FF3 compatible
Nim in JavaScript using GWT
The 21 Game in JavaScript
The hot game of Nim – Nim theory and connections with other games at cut-the-knot
Nim and 2-dimensional SuperNim at cut-the-knot
The Game of Nim at sputsoft.com
Nim on NCTM's Illuminations
Marienbad – Variant of Nim HTML5 application
Pick Stones Python Source Code of a Marienbad Variant of Nim using Alpha-Beta Pruning
NIM in Excel Excel spreadsheet using basic formulas to determine best moves for a 3,4,5 start point.
Play NIM online against computer opponents of different strength. Adjustable initial position.

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Nim