Larger, Smaller or Similar?

Which of the yellow areas is larger?

Here is comparison of areas! 

Both Sharing Equal Area!

Which areas are into comparison?

Actually, areas of both are equal. The diagonal divide the rectangle into 2 halves. So triangle A and A' or B and B' have equal areas.

When diagonal divides the area of main rectangle into 2 halves, area of triangles A (or A') and area of triangle B (or B') are further subtracted from each half to get the areas of the shaded region.

Since equal areas are subtracted from triangles formed by diagonal to get the shaded area, the area of shaded parts are equal. 

 That is from each half area subtracted = A + B = A' + B'.

The Domino on The Chessboard Challenge

There is an 8 by 8 chessboard in which two diagonally opposite corners have been cut off.You are given 31 dominos, and a single domino can cover exactly two squares. Can you use the 31 dominos to cover the entire board?

Simple Arrangement? Check out it's possibility!

Impossible Dominos' Arrangement on Chessboard

What was the challenge given?

Initial mathematical calculations might suggest that the task is pretty simple. If 2 square are cut off from 64 squares then 62 squares will be left which are enough for 31 dominos (each covering 2 squares).

But, that is not the case. Since, 2 diagonally opposite squares are removed, they has to be either black or white like shown below with shaded regions.

We need 1 black and 1 White square for placement of 1 domino on the chessboard.That is 31 Black and 31 White squares are needed to give cover for 31 dominos.

 In the above 2 cases, there are either 32 Black and 30 White or 30 Black and 32 White squares are available.

This makes the task of placing 31 dominos on the chessboard (whose 2 diagonally opposite squares are removed) impossible!