Subtraction games
Subtraction games
Ryan Cox, Donovan Di Dio
April 5, 2017
1/8
Subtraction games
Definitions
Combinatorial Game
Zermelo’s Theorem
Winning Strategy
P-position
N-position
2/8
Subtraction games
Definitions
Combinatorial Game
Zermelo’s Theorem
Winning Strategy
P-position
N-position
2/8
Subtraction games
Definitions
Combinatorial Game
Zermelo’s Theorem
Winning Strategy
P-position
N-position
2/8
Subtraction games
Definitions
Combinatorial Game
Zermelo’s Theorem
Winning Strategy
P-position
N-position
2/8
Subtraction games
Definitions
Combinatorial Game
Zermelo’s Theorem
Winning Strategy
P-position
N-position
2/8
Subtraction games
Definitions
Combinatorial Game
Zermelo’s Theorem
Winning Strategy
P-position
N-position
2/8
Subtraction games
Subtraction Game
Rules
Let N be a positive integer.
Let the set S include at least 1 positive integer
Players take turns subtracting an integer in S from N, giving a
new N for the next player’s turn.
N is not allowed to be negative.
Last player to move wins!
3/8
Subtraction games
Subtraction Game
Rules
Let N be a positive integer.
Let the set S include at least 1 positive integer
Players take turns subtracting an integer in S from N, giving a
new N for the next player’s turn.
N is not allowed to be negative.
Last player to move wins!
3/8
Subtraction games
Subtraction Game
Rules
Let N be a positive integer.
Let the set S include at least 1 positive integer
Players take turns subtracting an integer in S from N, giving a
new N for the next player’s turn.
N is not allowed to be negative.
Last player to move wins!
3/8
Subtraction games
Subtraction Game
Rules
Let N be a positive integer.
Let the set S include at least 1 positive integer
Players take turns subtracting an integer in S from N, giving a
new N for the next player’s turn.
N is not allowed to be negative.
Last player to move wins!
3/8
Subtraction games
Subtraction Game
Rules
Let N be a positive integer.
Let the set S include at least 1 positive integer
Players take turns subtracting an integer in S from N, giving a
new N for the next player’s turn.
N is not allowed to be negative.
Last player to move wins!
3/8
Subtraction games
Example 1
N=35
S= {2,3,5}
0
P-position
1
P-position
2
N
3
N
4
N
5
N
6
N
7
P
8
P
From N=6, it begins repeating the same string of 2 P-positions
and 5 N-positions.
With N=35, player TWO has a winning strategy.
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Subtraction games
Example 2
In some cases, it is obvious who has the winning strategy.
N=256
S={300, 500, 160000, 523}
In this example, Player TWO clearly has the winning strategy,
as player ONE is not able to make a move.
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Subtraction games
Prime subtraction
In a very interesting game, let S= {1,P} where P is the set of all
primes. In this game, you can subtract 1, 2, and 3 mod 4, but
not 0. Thereby, the winning strategy is to move it to a position
where N mod 4 = 0.
N
0 1 2 3 4 5 6 7 8 9 10 11
P/N position P N N N P N N N P N N N
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Subtraction games
Example 3
Also, some games can take an extraordinary amount of time.
N=16,000,000
S= {1,2}
This game will take at least 8,000,000 moves.
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Subtraction games
See you next year!
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