|
|
| Line 1: |
Line 1: |
| '''Wythoff's game''' is a two-player [[mathematics|mathematical]] [[game of strategy]], played with two piles of counters. Players take turns removing counters from one or both piles; in the latter case, the numbers of counters removed from each pile must be equal. The game ends when one person removes the last counter or counters, thus winning.
| | Hi there, I am Sophia. He is an order clerk and it's something he really enjoy. Ohio is where my house is but my husband desires us to transfer. To climb is something I really appreciate performing.<br><br>Here is my web page; best psychic ([http://koreanyelp.com/index.php?document_srl=1798&mid=SchoolNews investigate this site]) |
| | |
| [[Martin Gardner]] 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.
| |
| | |
| ==Optimal strategy==
| |
| Any position in the game can be described by a pair of integers (''n'', ''m'') with ''n'' ≤ ''m'', describing the size of both piles in the position. 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'' > 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), 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).
| |
| | |
| ==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</sub> 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}}
| |
| | |
| [[Category:Mathematical games]]
| |
Hi there, I am Sophia. He is an order clerk and it's something he really enjoy. Ohio is where my house is but my husband desires us to transfer. To climb is something I really appreciate performing.
Here is my web page; best psychic (investigate this site)