and why this is important?

Because originally we have 64 cells and 32 of them were black and 32 white.

They were just four in each,

black and four white in each row and in each column.

So, now we have,

if you use a tile,

we use always one black cell and one white cell.

So, if we have an equal number,

we have 32 white and on restore it to black.

So, two whites will remain because they don't have black pairs,

so, it's impossible to cover the board.

So, let's make the serious,

explain it like mathematician seriously,

pretend that we are serious mathematicians.

So, a theorem, a Chessboard 8*8 without

two opposite corner cannot be tiled by dominos 1*2.

Why?

The proof.

First, we note that in the coloring we have

black and white cells and in each row and each column we have four black and four white.

So, we can count the total number of cells.

And opposite corners are black,

they can be also white,

but let's consider the case when they are black.

And then we have 30 black and 32 white cells,

and this numbers are different each domino has two different colors,

so, it covers one black and one white.

And there is no chance that we cover everything at least two white should

remain and like mathematician like say ''quod erat demostradum''.

It's Latin which means that,

that's what we wanted to prove.

So, this is the prove.

Now, let's see what we can learn from these examples.

So, I hope that you are now convinced that indeed we cannot tile the board.

The proof can be really convincing hopefully.

And there are many questions we can ask after that,

imagine we cut two corners but they are not opposite corners,

but corners on one side.

Can we tell the rest,

they are actually of different colors.

So, the obstacle doesn't exist,

can you tile the rest?

Or imagine we just cut any two cells of different colors,

is it always possible to tile the rest?

And I will not tell you the solution,

but I will just show you some picture which contains some ideas.

So, this picture is a bit strange but still it's look like this.

So, why the snake?

There is a snake going like.

So, why the snake?

What is the connection of the snake with our problem.

So, we cut two cells of different colors, black and white.

And we want to tile the rest with the dominos,

how the snake can help us, do you see this.