d038: Nim Game (Again)
Po-Lung Chen
Team Dont Block Me, National Taiwan University
March 26, 2010
Po-Lung Chen (Dont block me)
d038: Nim Game (Again)
2010/03/26
(1)
Nim Game
• The Nim Game consists of some piles of stones, and
the players take turn to grab stones from some pile(s).
• The player who grabs the last stone loses the game.
pile1
Po-Lung Chen (Dont block me)
pile2
d038: Nim Game (Again)
pile3
2010/03/26
(2)
Problem Description (1/3)
• A Nim Game variant:
– 2 piles of stones, each pile consists of zero or more stones.
– 2 players, alternately grab stones from the pile(s).
– The one who grabs the last stone loses the game.
• Grabbing rule:
– Grab at least one stone from one of the two piles.
– Grab from each of the two piles the same number of stones.
Po-Lung Chen (Dont block me)
d038: Nim Game (Again)
2010/03/26
(3)
Problem Description (2/3)
• The player who does NOT grab first will have one
chance of launching a “counterattack” during the
game play.
• Counterattack:
Po-Lung Chen (Dont block me)
d038: Nim Game (Again)
2010/03/26
(4)
Problem Description (3/3)
• Two experts playing this new Nim game.
• Their strategies are always the best possible.
– If the current situation would lead to a win:
• choose the way minimizing #steps.
– If the current situation would lead to a loss:
• choose the way maximizing #steps.
• Given the number of the stones
• Determine:
and
,
– (1) First player will win or lose.
– (2) The total number of steps in this game.
Po-Lung Chen (Dont block me)
d038: Nim Game (Again)
2010/03/26
(5)
Solution (1/4)
• This problem can be solved by dynamic programming.
• A “state” of the game consists of:
–
–
–
–
# of stones in first pile
# of stones in second pile
the current player has the chance to use counterattack?
the other player has the chance to use counterattack?
• A win/lose function can be determined by the “state”.
•
Po-Lung Chen (Dont block me)
can be “WIN” or “LOSS”
d038: Nim Game (Again)
2010/03/26
(6)
Solution (2/4)
• The recurrence relation:
–
“WIN”, if
•
•
•
•
•
– Otherwise
“LOSS”
• The initial condition:
–
,
and
Po-Lung Chen (Dont block me)
,
are all “WIN” states.
d038: Nim Game (Again)
2010/03/26
(7)
Solution (3/4)
• Similarly, we can define the function for the number of
steps in game according to players’ strategies.
• Case 1 (the state is going to win):
Find minimum
(if possible)
Po-Lung Chen (Dont block me)
d038: Nim Game (Again)
2010/03/26
(8)
Solution (4/4)
• Case 2 (the state is going to lose):
Find maximum
(if possible)
• For initial conditions, set them all to zero.
Po-Lung Chen (Dont block me)
d038: Nim Game (Again)
2010/03/26
(9)
Time Complexity
•
•
•
•
Let be the maximum number of stones +1 in a pile.
The number of possible states is no more than
.
Calculation time for a state is
.
Therefore the time complexity is
.
• We can use pre-calculation to find out the answer to
all the possible states.
Po-Lung Chen (Dont block me)
d038: Nim Game (Again)
2010/03/26
(10)
Implementation Details
• Be careful! The maximum number of stones in a pile
can be 110, not 100. This might cause runtime error if
the array is too small.
Po-Lung Chen (Dont block me)
d038: Nim Game (Again)
2010/03/26
(11)
Finally…
Thanks for your attention!
Po-Lung Chen (Dont block me)
d038: Nim Game (Again)
2010/03/26
(12)