Transcript Princess and the Roses - University of California, Irvine

Princess and the
Roses
Naomi Carrillo &
Samantha Helstrom
iCamp 2012
Background
• Questions about Princess and the Roses are
unsolved
• Featured in Winning Ways for Your
Mathematical Plays
– Small number of bushes with large number of
roses: “parity considerations are paramount”
– Large number of bushes with few roses: “triality
triumphs”
• Concept- two princes fighting for the love of
a princess by bringing her roses
Game Description
• 2 player game
• n number of bushes with any number of roses in
each bush
• Notation- array of numbers
Ex: [6,3,7,4,2]
• Legal move: remove ONE rose from ONE bush
or
remove ONE rose from TWO different bushes
• Person to remove last rose wins
Game Example
1
2
3
Player 1 removed 2 roses from bush 1 and
bush 2
1
2
3
Player 2 removed 2 roses from bush 2 and
bush 3
1
2
3
Player 1 removed 1 rose from bush 2
1
2
3
Player 2 removed 2 roses from bush 1 and
bush 2
1
2
3
Player 1 removed 2 roses from bush 1 and
bush 3
1
2
3
Game classification
• Determinate: win does not rely on chance
• Zero-sum: one definite winner and loser
• Symmetric: payoff of game depends on
what strategy the player chooses
• Perfect information: everything is known
by both players
• Sequential: players alternate turns
• Normal: person to take last rose wins
Questions for Investigation
• Is the game fair or unfair?
• Which player has the winning strategy for
certain game states?
• How many different game states exist for
a starting game board?
Combinatorics
• How many game states?
m = # of bushes
ni = # of roses
n1 . n2 . n3 . . . nm-1 . nm
Multiply the number of roses in each bush
by each other.
Results for 1 bush game
# of roses
position
1
n
2
p
3
n
4
p
5
n
The pattern continues…
n- next player has
winning strategy
p- previous player
has winning
strategy
Results for 2 bush game
# of roses
position
(odd,odd)
n
(odd,even)
n
(even,even)
p
n- next player has
winning strategy
p- previous player
has winning
strategy
winning strategy- make # of roses in both
bushes EVEN on your turn
Game Tree Example
(2,3)
(2,2)
(1,2)
(1,1)
(0,1) (0,2) (1,1) (1,0)
(0,0)
(0,1) (0,0) (0,1) (0,0)
(0,0)
(0,0)
(0,0)
(0,2)
(1,3)
(1,2)
(1,2)
(0,3) (0,1) (0,2)
(0,1) (0,1) (0,2) (1,1)
(0,0)
(0,2)
(0,0) (0,1) (0,1)(0,0) (0,1)
(0,0) (0,0)
(0,0)
(0,0)
(1,1)
(0,1) (0,0)
(0,0)
(1,0)
(0,0)
Results for 3 bush game
# of roses
position
(odd,odd,odd)
p
(odd,even,even)
n
(odd,odd,even)
n
(even,even,even)
p
n- next player has
winning strategy
p- previous player
has winning
strategy
winning strategy- make # of roses in all 3 bushes
either ALL odd or ALL even on your turn
Adaptive Learning Results
(2,2,2)
Player 2 has winning strategy
Adaptive Learning Results
(3,3,2)
Player 1 has winning strategy
Overall Results
• Game is UNFAIR
• Winning strategy depends on oddness
and evenness of roses in each bush
• Adaptive learning program can determine
which player has winning strategy for any
starting game board (as # of bushes and
roses increases, the program takes longer)
Future Work
• Find winning strategy for a game of 4
bushes, 5 bushes, 6 bushes, etc.
• Continue to work with the adaptive
learning program
Any Questions?
Thank
you!