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

 

  1. Pick at least one stone
     
  2. Pick any number of stones from one of the piles.
     
  3. 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
nA
n \notin A
nA

b(n+1) 

b(n) + 2

b(n) + 3

n \in B
nB
n \notin B
nB

a(1) = 1

b(1) = 2

Algorithm Complexity

O (m*n)

 

Vs

 

O(max(m,n))

Game formats

Vs

Vs

That's All Folks!

  1. Deeksha Tandon
  2. Md. Zurez Tuba
  3. Shreya Khurana
  4. Shubham Goel
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