Minimax Approach to Wythoff's Game
Minimax Algorithm
And I'm Min!
Hi!
I'm Max.
Minimax Algorithm
Oh!
You wish.
I'm going to take you down.
Minimax Algorithm
As if that's going to help. I'll choose min.
Well then. I'm going to use Max at each of my steps.
Minimax Algorithm
Max wants to maximize its outcome
Min wants to minimize Max's outcome
Game seen from Max's point of view
Wythoff's Game
Wythoff's Game
Rules
-
Pick at least one stone
-
Pick any number of stones from one of the piles.
- Pick same number of stones from both the piles.
I'll just pick up all stones at my first chance.
Cheater.
Is this game solvable?
Yes.
Solving the Game
Grid method
Properties of Wythoff pairs
Game tree
So we decided to change it.
Game tree
Position Grid
Putting the Why in Wythoff
Putting the Why in Wythoff
Putting the Why in Wythoff
Err..why the name position grid?
Losing positions
Winning positions
Properties of Safe Pairs
a(n) = [n ] and
b(n) = [n ] = a(n) + n
\alpha
α
\alpha^2
α2
\alpha
α
= Golden Section Ratio =
[n] is the greatest integer not exceeding n.
(1 +\sqrt{5})/ 2
(1+5)/2
Properties of Safe Pairs
a(n+1)
a(n) + 2
a(n) + 1
n \in A
n∈A
n \notin A
n∈/A
b(n+1)
b(n) + 2
b(n) + 3
n \in B
n∈B
n \notin B
n∈/B
a(1) = 1
b(1) = 2
Algorithm Complexity
O (m*n)
Vs
O(max(m,n))
Game formats
Vs
Vs
That's All Folks!
- Deeksha Tandon
- Md. Zurez Tuba
- Shreya Khurana
- Shubham Goel
Minimax Approach to Wythoff's Game