If each small square on a grand chess board is 2 cm by 2 cm, each diagonal

of the whole board measures 800 cm. How many small squares are on the
board?

To find the number of small squares on the chess board, we need to determine the dimensions of the board first. Since each diagonal of the whole board measures 800 cm, we can find the length of each side of the board.

Let's assume the length of each side of the board is L cm.

Using the Pythagorean theorem, we know that the square of the hypotenuse (diagonal) of a right triangle is equal to the sum of the squares of its two legs (sides).

We can set up the equation as follows:

L^2 + L^2 = 800^2

2L^2 = 640,000

Dividing both sides by 2:

L^2 = 320,000

Taking the square root of both sides:

L ≈ √320,000

L ≈ 565.685 cm

Since each small square on the board is 2 cm by 2 cm, we can determine the number of squares on each side of the board by dividing the length of the side by the length of each square:

Number of squares on each side = L / 2

Number of squares on each side = 565.685 / 2

Number of squares on each side ≈ 282.843 squares

To find the total number of squares on the board, we multiply the number of squares on each side by itself:

Total number of squares = (Number of squares on each side) ^ 2

Total number of squares ≈ 282.843 ^ 2

Total number of squares ≈ 79,847.750 square

Since we can't have a fraction of a square, we round down the final answer.

Therefore, there are approximately 79,847 small squares on the chess board.

To find the number of small squares on the board, we need to determine the dimensions of the grand chessboard first.

Let's assume that the grand chessboard has X number of small squares along the horizontal side and Y number of small squares along the vertical side.

Given that each small square on the board is 2 cm by 2 cm, the distance covered by each diagonal of the board is 800 cm.

Since the diagonal traverses the entire board, we can find its length using the Pythagorean theorem:

Diagonal² = (X * 2 cm)² + (Y * 2 cm)²

Simplifying this equation, we get:

640,000 cm² = 4 cm² * (X² + Y²)

Dividing both sides by 4 cm², we have:

160,000 cm² = X² + Y²

The number 160,000 can be represented as the sum of two squares in several ways, but since we are looking for dimensions that correspond to a chessboard, we need to express it as a sum of two square numbers that have the same parity (both even or both odd).

The only way to represent 160,000 in this form is as 400² + 0².

Therefore, X² = 400² and Y² = 0².

Taking the square root of both sides, we find:

X = 400 and Y = 0

Given that Y = 0, it means that the grand chessboard has no vertical squares. Therefore, there are only horizontal squares.

So, the number of small squares on the board is simply X, which is 400.

Hence, there are 400 small squares on the board.

?? The sides appear not to be integers. I guess there must be a border included in the length.

√800 = 20√2

So, I guess there are 10√2 small squares across and down.

If they were 1 by 1 the diagonal would be 400

say n of them along each side
diagonal is then n sqrt 2 =400
n = 400/sqrt 2
but n^2 is number of squares
so
answer = 400^2/2 =200*400 = 80,000
I suspect you mean the diagonal is 80 cm, not 800 by the way