View source for Wythoff's game

{{Short description|Two-player mathematical subtraction game}}
{{pp-sock|small=yes}}
[[File:Westbury sub Mendip 2016T462. 189 silver denarii cleaned for ID, January 2017. Two piles as hoard divided by find days. (FindID 810901).jpg|thumb|Wythoff's game is played with two piles of counters]]
'''Wythoff's game''' is a two-player mathematical [[subtraction game]], played with two piles of counters. Players take turns removing counters from one or both piles; when removing counters from both piles, the numbers of counters removed from each pile must be equal. The game ends when one player removes the last counter or counters, thus winning.

An equivalent description of the game is that a single [[Queen (chess)|chess queen]] is placed somewhere on a large grid of squares, and each player can move the queen towards the lower left corner of the grid: south, west, or southwest, any number of steps. The winner is the player who moves the queen into the corner. The two [[Cartesian coordinates]] of the queen correspond to the sizes of two piles in the formulation of the game involving removing counters from piles.

[[Martin Gardner]] in his March 1977 "[[Mathematical Games column]]" in ''[[Scientific American]]'' claims that the game was played in China under the name 捡石子 ''jiǎn shízǐ'' ("picking stones").<ref>[http://www.cut-the-knot.org/pythagoras/withoff.shtml Wythoff's game at Cut-the-knot], quoting [[Martin Gardner]]'s book ''[[Penrose Tiles to Trapdoor Ciphers]]''</ref> The Dutch mathematician [[Willem Abraham Wythoff|W. A. Wythoff]] published a mathematical analysis of the game in 1907.<ref>{{citation
 | last = Wythoff | first = W. A. | author-link = Willem Abraham Wythoff
 | issue = 2
 | journal = Nieuw Archief voor Wiskunde
 | pages = 199–202
 | title = A modification of the game of nim
 | url = https://archive.org/details/nieuwarchiefvoo02genogoog/page/n219/mode/2up
 | volume = 7
 | year = 1907}}</ref>

==Optimal strategy==
[[File:Wythoff's Nim.svg|thumb|upright=1.6|A visualization of Wythoff's game of Nim. The bottom leftmost square is position (1,1) and the red 
squares are cold positions. Note that the winning square is not included in the picture.]]
Any position in the game can be described by a pair of [[integers]] (''n'', ''m'') with ''n''&nbsp;≤&nbsp;''m'', describing the size of both piles in the position or the coordinates of the queen. The strategy of the game revolves around ''cold positions'' and ''hot positions'': in a cold position, the player whose turn it is to move will lose with best play, while in a hot position, the player whose turn it is to move will win with best play. The [[optimal]] strategy from a hot position is to move to any reachable cold position.

The classification of positions into hot and cold can be carried out [[recursion|recursively]] with the following three rules:
#(0,0) is a cold position.
#Any position from which a cold position can be reached in a single move is a hot position.
#If every move leads to a hot position, then a position is cold.

For instance, all positions of the form  (0, ''m'') and (''m'', ''m'') with ''m''&nbsp;>&nbsp;0 are hot, by rule 2. However, the position (1,2) is cold, because the only positions that can be reached from it, (0,1), (0,2), (1,0) and (1,1), are all hot. The cold positions (''n'', ''m'') with the smallest values of ''n'' and ''m'' are (0, 0), (1, 2), (3, 5), (4, 7), (6, 10) and (8, 13). (sequence {{OEIS link|A066096}} and {{OEIS link|A090909}} in [[OEIS]]) (Also see {{oeis|A072061}})

For [[Misère|misère game]] of this game, (0, 1) and (2, 2) are cold positions, and a position (''n'', ''m'') with ''m'',&nbsp;''n''&nbsp;>&nbsp;2 is cold if and only if (''n'', ''m'') in normal game is cold.

==Formula for cold positions==
Wythoff discovered that the cold positions follow a regular pattern determined by the [[golden ratio]]. Specifically, if ''k'' is any natural number and 
:<math>n_k = \lfloor k \phi \rfloor = \lfloor m_k \phi \rfloor -m_k \,</math>
:<math>m_k = \lfloor k \phi^2 \rfloor = \lceil n_k \phi \rceil = n_k + k \,</math>
where φ is the golden ratio and we are using the [[floor function]], then (''n''<sub>''k''</sub>, ''m''<sub>''k''</sub>) is the ''k''<sup>th</sup> cold position. These two sequences of numbers are recorded in the [[Online Encyclopedia of Integer Sequences]] as {{OEIS2C|id=A000201}} and  {{OEIS2C|id=A001950}}, respectively.

The two sequences ''n<sub>k</sub>'' and ''m<sub>k</sub>'' are the [[Beatty sequence]]s associated with the equation
:<math>\frac{1}{\phi} + \frac{1}{\phi^2} = 1 \,.</math>
As is true in general for pairs of Beatty sequences, these two sequences are [[complement (set theory)|complementary]]: each positive integer appears exactly once in either sequence.

==See also==
* [[Nim]]
* [[Grundy's game]]
* [[Subtract a square]]
* [[Wythoff array]]

==References==
{{reflist}}

==External links==
*{{MathWorld|urlname=WythoffsGame|title=Wythoff's Game}}
* {{cite web|last1=Grime|first1=James|title=Wythoff's Game (Get Home)|url=https://www.youtube.com/watch?v=AYOB-6wyK_I |archive-url=https://ghostarchive.org/varchive/youtube/20211215/AYOB-6wyK_I |archive-date=2021-12-15 |url-status=live|website=YouTube|publisher=[[Brady Haran]]|accessdate=21 August 2017|format=video}}{{cbignore}}

[[Category:Mathematical games]]