Chess and Math

There are deep relation between chess and math. But unfortunately we do less works on it. There are several pure research which are not publicly available. We common people can help each other to share such kind of knowledge.

[ Update of important Posts: 2,3,14,19,35,52, 84 ...]

Board in Equation:

we know chessboard is in 2D kartesian system. sothere are geometrical relations are available there.

Lets express the Vertical, Horizontal and Diagonal lines.

Vertical lines:

a1 to a8 = (1,1) to (1,8) : X=1

b1 to b8 = (2,1) to (2,8): X=2

c1 to c8 = (3,1) to (3,8) : X=3

d1 to d8 = (4,1) to (4,8) : X=4

e1 to e8 = (5,1) to (5,8) : X=5

f1 to f8 = (6,1) to  (6,8) :X=6

g1 to g8 = (7,1) to (7,8) :X=7

h1 to h8 = (8,1) to (8,8) :X=8

Horizontal lines:

a1 to h1: Y=1

a2 to h2: Y=2

a3 to h3: Y=3

a4 to h4: Y=4

a5 to h5: Y=5

a6 to h6: Y=6

a7 to h7: Y=7

a8 to h8: Y=8

Diagonals:

positively sloped:

g1 to h2: X-Y=6

f1 to h3: X-Y=5

e1 to h4: X-Y=4

d1 to h5: X-Y=3

c1 to h6: X-Y=2

b1 to h7: X-Y=1

a1 to h8: X-Y=0

a2 to g8: X-Y=-1

a3 to f8: X-Y=-2

a4 to e8: X-Y=-3

a5 to d8: X-Y=-4

a6 to c8: X-Y=-5

a7 to b8: X-Y=-6

Negatively Sloped:

b1 to a2: X+Y=3

c1 to a3: X+Y=4

d1 to a4: X+Y=5

e1 to a5: X+Y=6

f1 to  a6: X+Y=7

g1 to a7: X+Y=8

h1 to a8: X+Y=9

h2 to b8: X+Y=10

h3 to c8: X+Y=11

h4 to d8: X+Y=12

h5 to e8: X+Y=13

h6 to f8: X+Y=14

h7 to g8: X+Y=15

Light and Dark Squares:

Choose a square d4
-> convert it in co-ordination (4,4)
-> sum of co-ordination  [4+4=8]
-> It is an even number
-> It is a Dark Square

Again ,

Choose a Square e4
-> Convert it as (5,4)
-> Sum it [5+4=9]
-> It is an odd number
->It is a light Square

So, Sum of Co-ordination tells the color of Squares.

A bit easy I'm afraid.  Have you read Hawking's book "God created the integers"?  It has awesome stuff in it.  Euclid, Newton, differential calculus, awesome!  By the way, what grade are you in?

In which point of view ?

Beautiful, but really easy calculations.  That's what I mean.

Extra-ordinary works seem easy after it is done.

It's not extraordinary.  Euclid's theorems are extraordinary.

I will share more fantastic works, it is just Basic. Yet original door is not opened.

Don't worry, so will I.  Although I'm more focused on the geometry of the Multiverse.

If my works are not extra ordinary , i should stop sharing more information.

Honestly, they are unique, which makes them in a way extra-ordinary, but they are not extra-ordinary in terms of the brainpower it took to come up with them.  I do appreciate them as a mathematician and comrade however.

Thank you and wait for more surprise, give me a day to type.

Now i have to sleep.

Chess in Set

M = set of materials
   = { W, B}

W = set of white materials
    = {P,N,B,R,Q,K}
B  = set of Black Materials
    = {P',N',B',R',Q',K'}

S = Set of all squares in Board
   = { a1,a2,a3,......h8}

n(S)=64

S= {Ls,Ds}

Ls = Set of Light squares

[ if, S(x,y) is a square and (x+y) belong to N(2n+1) where N is  set of positive odd integer, then S(x,y) belong to Ls]

Ds = Set of Dark Squares

[if, S(x,y) is a square and (x+y) belong to N(2n) where N is  set of positive even integer, then S(x,y) belong to Ds]

 # Based on Grabbed in and checked in , The Squares are classified into 2 classes-

• Grabbed Squares
• Empty Squares

* Grabbed Squares are two types -

• White grab
• Black grab

*Empty Squares are two types-

• Checked Squares [holding/controlling]
• Non Checked Squares

very nice a big salute to you

very nice a big salute to you

Equation for Knight route

Suppose Knight stays on d4, 

  so, Nd4 holds the Squares e6,c6,b5,b3,c2,e2,f3,f5

 if we use "hold" with "x"  Sign and Grab with "x" sign,it should be expressed as

       N x d4      x {e6,c6,b5,b3,c2,e2,f3,f5 }
or,   N x (4,4)   x {e6,c6,b5,b3,c2,e2,f3,f5 }
or,   N(4,4)       x  {(5,6),(3,6),(2,5),(2,3),(3,2),(5,2),(6,3),(6,5)}

 The Diagram of knight route is nothing but a circle. and (4,4) is the centre of circle.

According circle equation, the equation of knight route:

     (x-4)^2+(y-2)^2= ( root square(5))^2
or, (x-4)^2+(y-2)^2= 5

in general form,

(x-a)^2+(y-b)^2= 5  [ (a,b) the square of knight]

or, (x^2+ y^2) + (a^2+b^2) = 5 + 2(ax+by)

if so , you should post it before i post in forum.

That is just basic. yet there are many thing to show.