Y2K game

from Wikipedia, the free encyclopedia

The Y2K game (English Y2K Game or SOS Game ) is a 2-person game that can be played with paper and pencil and which formed the basis of a task at the US Mathematical Olympiad in 1999 . It has since been included in several mathematical publications.

Y2K (English Year 2 Kilos ) stands for the year 2000 . The authors have formulated the rules of the game in such a way that the number 2000 plays a role in it. This is the only connection between the game and Y2K, as the game allows any other number values. The admission to a mathematics Olympiad took place because of the obvious problem of finding a winning strategy .

Rules of the game

Numbered sequence of moves (example)
Player A blue, player B black, B wins on move 8

The game board consists of 2000 squares arranged next to each other (initially empty). Players A (attracting) and B (trailing) alternately fill in the fields in any order; only the entries S or O are allowed. The player who creates the partial sequence ... SOS ... first is the winner.

Game theory classification

The Y2K game is a finite, random and neutral (or objective ) zero-sum game with complete and perfect information. According to Ernst Zermelo's theorem , in a finite, random, zero-sum game with perfect information, there is either a winning strategy for A, or there is a winning strategy for B, or the game ends in a draw if both players play optimally. In the mathematical literature it is proven for the Y2K game that B has a winning strategy, i.e. always wins if he does not make a mistake.

The role of the number 2000

If you take the Y2K game as a real game and not just a mathematical problem, it quickly becomes clear that 2000 fields are very unwieldy and lead to a long game time. The analysis of the winning strategy shows that z. B. 20 fields would be sufficient. It can be assumed that the number 2000 was chosen in order to be able to give the game an attractive name in connection with the upcoming millennium in the year of publication in 1999 .

Expansion of the game idea

The game becomes more challenging when different numbers of playing fields are allowed. Then it is also possible that player A has a chance of winning or the game ends in a draw if A and B both play optimally. It can be shown that there is a winning strategy for A if the number of fields is at least 7 and odd. There is a winning strategy for B if the number of squares is at least 16 and even. In all other cases there is no winning strategy.

literature

T. Andreescu, Z. Feng (Ed.): Mathematical Olympiads 1998-1999: Problems and Solutions From Around the World . Mathematical Association of America 2000, ISBN 978-0883858035

Web links

Individual evidence

  1. ↑ 1999 USAMO Problems (en.) . Website Art of Problem Solving . Retrieved June 11, 2015.
  2. ^ P. Winkler: Mathematical Mind-Benders . AK Peters 2007, ISBN 978-1568813363 , p. 76
  3. ↑ 29th Annual USA Mathematical Olympiad Problems and Solutions . Mathematics Magazine 73/3 (2000), pp. 248-252
  4. ↑ a b K. Y. Li: Mathematical Games II (en.) ( Memento of February 4, 2012 in the Internet Archive ). Website Mathematical Excalibur 7/5 (2003), University of Hong Kong. Retrieved March 29, 2015.
  5. ^ TS Ferguson: Game Theory (en.) . P. I-8. Retrieved March 29, 2015.
  6. ↑ Jörg Bewersdorff : Luck, Logic and Bluff: Mathematics in Play - Methods, Results and Limits , Vieweg + Teubner , 5th edition 2010, ISBN 3834807753 , doi: 10.1007 / 978-3-8348-9696-4 , p. 94– 102
  7. ↑ a b P. Winkler: Mathematical Mind-Benders . AK Peters 2007, ISBN 978-1568813363 , p. 78
  8. ↑ a b M. Börgens: 101 . Website Math Problems # 76 (2011). Retrieved June 11, 2015.