Mathematical Games

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Mathematical Games

• Charlie Turner
• February 2006

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What is a mathematical game?

• game
• An activity providing entertainment or
  amusement a pastime party games word games.
• (Mathematics) A model of a competitive
  situation that identifies interested parties and
  stipulates rules governing all aspects of the
  competition, used in game theory to determine the
  optimal course of action for an interested party.
  

from www.dictionary.com
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Why play mathematical games?
Intellectual challenge
links between seemingly different games
a different way of explaining
9 card game, jam and hot
motivate learning
fun!
discover new underlying mathematics
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Games on Graphs
In his book On Numbers and Games John Conway
shows how games can be used to describe numbers.
Kayles
Rim
Rayles
Hackenbush breaking up pictures made of spots and
lines
John Conway!
Sprouts linking spots and lines
Nim
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Sprouts the rules
Start with n dots on a piece of paper A move
consists of drawing a line and adding a dot
anywhere along the line Restrictions The line
may not cross itself or any previously made line
or pass through any previously made spot. No spot
may have more than 3 lines emanating from
it. Players take turns we will play that the
first person unable to play loses (so last person
to play wins!)
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Sprouts analysis
Every sprouts game is finite!
Think in terms of lives
Idea of strategy - to control number of live
spots left at the end
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(childish) Hackenbush (unrestrained)

• Picture is made of jointed lines, with dots.
• Every part of the picture is connected to the
  frame
• Players take it in turns to remove lines
• If you remove a line, then all other parts that
  are now disconnected from the frame are also
  removed
• The person who removes the last part of the
  picture wins
• Related to nim
• An Impartial game

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Hackenbush (unrestrained)
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Hackenbush (restrained)
Hackenbush (restrained) also called blue-red
hackenbush. No longer impartial Sometimes more
complicated because your opponent can control
your lines
Blue-red Hackenbush positions lead to the surreal
numbers Invented by John Conway in 1969 Surreal
numbers are the most natural collection of
numbers which includes both the real numbers and
the infinite ordinal numbers of Georg Cantor.
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Nim the takeaway game
Simple game, start with piles of counters (or
any other object) Usually played as a misère
game Can take any number of counters from a pile
in each move Totally solved for any number of
piles and objects Winning moves based on binary
digital sum of the heap sizes
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Nim - example
Add 101 (5) 100 (4) 011 (3) 010
(2) Then find the nim-sum of 2 with each pile
to find which pile is hence reduced, so reduce it
to the number found in the sum. 101 (5) 100
(4) 11 (3) 10 (2) 10 (2) 10
(2) 111 (7) 110 (6) 01 (1)
Nimbers and the Sprague-Grundy Theorem
Isomorphic to simple games of hackenbush
(unrestrained)
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Mathematical Games in the classroom
simple
investigation
Linking ideas together
Fun!
New areas of mathematics
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Extras!
Kayles
Rim
Rayles