Chessboard squares

Let's say that we add diagonally opposite squares... $64+1=65$ , $63+2=65$ , $62+3=65$ , $61+4=65$ , and so on and so forth... All diagonally opposite squares have the same sum. And since there's an equal number of white and yellow squares, we can say that the answer is $\boxed{0}$ .

Take any small 2x2 square and do as indicated the result is zero, this pattern persist in the whole board. So dividing our 8x8 board into 16. 2x2 miniboard we come out with zero too.

Of course, I did it a harder way by noticing a column equals 32 or -32 depending on the colour that starts. I only had to add up one column because each square as you go down is 8 more than the previous square. That means if you start from the top, the 2nd colour will have 8 more per square. of that colour. Since there are four of each colour in a column, the 2nd colour in each column will have 8x4 value more than the colour that starts.

Python:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16

white = 0
yellow = 0
current = 'y'
for n in xrange(1,65):
    if current == 'w':
        white += n
        current = 'y'
    else:
        yellow += n
        current = 'w'
    if n % 8 == 0:
        if current == 'w':
            current = 'y'
        else:
            current = 'w'
print "Answer:", yellow - white

easy, we gonna subtract the yellow square to the white square

in the first row :

$2-1=1, 4-3=1, 6-5=1, 8-7=1$

for the second row :

$9-10=-1, 11-12=-1, 13-14=-1, 15-16=-1$

so, we got 4 in the the even-numbered row and -4 in the odds-numbered row

and since there's equal even-numbered and odds-numbered row, so the answer is $\boxed{0}$

WE Added fist two rows white squares and also the yellow squares and subtracting the sums,it is ZERO.Same will apply for other rows as well. Therefore answer is ZERO.

It is a 8 * 8 Square Matrix, , We can think of it as Sum of sixteen 2 * 2 square matrices,, Now in each 2 * 2 matrix at top row yellow wins with 1 point, and at bottom row white wins with 1 points,,, Ultimately no one wins, the game ties.

Lets add the numbers seperaratly so hat it would add to 65. Since there is the same amount of white and black squares, the answer is 0.

Count the yellow squares. The answer should be 32. Then count the white squares. The answer also should be 32.

Then do the simple math. The answer is 0.

I just add them all up and then subtracted.

Sum(yellow squares-white squares) of first row from top is =4, Sum(yellow squares-white squares) of second row from top is=-4, Sum(yellow squares-white squares) of 1st row and 2nd row fro top=4-4=0.

Similarly, Sum(yellow squares-white squares) of 3rd and 4th row from top=0,Sum(yellow squares-white squares) of 5th and 6th row from top=0, & Sum(yellow squares-white squares) of 7th and 8th row from top=0. Therefore the answer is '0' (zero)

Half are white, half are brown. There are 32 of each. Thus subtracting them gives you zero.

Using Summation for Arithmetic series:

Sum of White boxes= 1/2 * 32 * (1 + 64) = 1040

Sum of Yellow boxes = 1/2 * 32 * ( 2 + 63) = 1040

Hence 1040 - 1040 = 0 (ANS)