Turning Turtles

At each move a player chooses an "O" and turns it into an "X". At the same time this player may, if he so wishes, change a letter in any other square to the left from the first one. In the left square, the player is allowed to turn "O" into "X" and also "X" into "O". To perform a move, the player should first click under the square he plans to change. After selecting 1 or 2 squares, click on the "Make Move" button. If you plan to change a single square (an "O" into an "X") you may click on that square directly. This is kind of a shortcut with which one should be cautious. You can't undo your moves. The player to make the last move wins.

In the original version, players turn turtles upside down and back to their feet. Not having any aptitude for painting, I settled on the TicTacToe symbols to present the two possible states of each square.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


Buy this applet
What if applet does not run?

The game is a Nim in disguise, Nim with eleven rows of lengths 1, 2, 3, ..., and 11. Each square corresponds to one row. "X" in a row indicates that a row is empty while "O" means that the row is full of check marks. Changing "O" into "X" is the same as removing a whole row of check marks. If you wish to remove m check marks from a row that has n > m of them, change the square number n and then change the square number n-m to the left. It does not matter whether at this point m contains "O" or "X".

Reference

  1. E. R. Berlekamp, J. H. Conway, R. K. Guy, Winning Ways for Your Mathematical Plays, v1, A K Peters, 2001.
  2. E. R. Berlekamp, J. H. Conway, R. K. Guy, Winning Ways for Your Mathematical Plays, v2, Academic Press, 1982.
  3. R. K. Guy, Fair Game, Comap's Explorations in Mathematics, 1989

Related material
Read more...

  • What Is a Combinatorial Game?
  • A Game of Candy Squares
  • A Sticky Problem
  • Another Sticky Problem
  • Date Game
  • Dawson's Chess: an Interactive Gizmo
  • Dawson's Kayles: an Interactive Gizmo
  • Grundy's Game
  • Hex 7
  • Kayles
  • Nimble: an Interactive Gizmo
  • Northcott's game (An Interactive Gizmo)
  • Odd Scoring
  • One Pile: an Interactive Gizmo
  • Plainim (An Interactive Gizmo)
  • Plainim Misere (An Interactive Gizmo)
  • Scoring: the simplest of the impartial games
  • Scoring Misere: an Interactive Gizmo
  • Scoring Misere: Two Heaps Perfect Strategy
  • The Fraction Game
  • The Silver Dollar Game
  • Silver Dollar Game With No Silver Dollar
  • Subtraction Game
  • TacTix: an Interactive Gizmo
  • Take-Away Games
  • Wythoff's Nim, Literal Implementation
  • Wythoff's Nim (An Interactive Gizmo)
  • |Contact| |Front page| |Contents| |Games| |Store|

    Copyright © 1996-2012 Alexander Bogomolny

     40601326

    A math books store at a unique math study site. Shopping at the store helps maintain the site. Thank you.
    Sites for teachers
    Sites for parents
    Terms of use
    Awards
    Interactive Activities

    CTK Exchange
    CTK Wiki Math
    CTK Insights - a blog
    Math Help
    Games & Puzzles
    What Is What
    Arithmetic
    Algebra
    Geometry
    Probability
    Outline Mathematics
    Make an Identity
    Book Reviews
    Stories for Young
    Eye Opener
    Analog Gadgets
    Inventor's Paradox
    Did you know?...
    Proofs
    Math as Language
    Things Impossible
    Visual Illusions
    My Logo
    Math Poll
    Cut The Knot!
    MSET99 Talk
    Old and nice bookstore
    Other Math sites
    Front Page
    Movie shortcuts
    Personal info
    Privacy Policy

    Guest book
    News sites

    Recommend this site

    Sites for parents

    Education & Parenting

    Search:
    Keywords:

    Google
    Web CTK
    Supported by
    3wVentures