# [SOLVED]Induction: Each square can be covered by L-stones

#### mathmari

##### Well-known member
MHB Site Helper
Hey!!

A square with the side length $2^n$ length units (LU) is divided in sub-squares with the side length $1$. One of the sub-squares in the corners has been removed. All other sub-squares should now be covered completely and without overlapping with L-stones. An L-stone consists of three sub-squares that together form an L.

I want to draw the problem for the first three cases described above ($1 \leq n \leq 3$).

Then I want to show the following using induction:

For all $n \in N$ the square with side length $2^n$ LU can be covered completely and without overlapping with L-stones, after one of the sub-squares in the corners has been removed.

For the first part:

Is the drawing correct?

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#### mathmari

##### Well-known member
MHB Site Helper
Can we use the sketch of the case $n=2$ to get the one of the case $n=3$ ?

Is it maybe as follows?

The upper right sub-square is the one of case $n=2$. For the other sub-squares we have to fill them completely.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Can we use the sketch of the case $n=2$ to get the one of the case $n=3$ ?

Is it maybe as follows?

The upper right sub-square is the one of case $n=2$. For the other sub-squares we have to fill them completely.
Hey mathmari !!

I think so yes.
Suppose we use the same case $n=2$ square to fill each of the 4 sub squares of the case $n=3$.
Then we have 3 cells left that we still have to fill don't we?
Can we align them so that we can put another L-square into it?

#### mathmari

##### Well-known member
MHB Site Helper
I think so yes.
Suppose we use the same case $n=2$ square to fill each of the 4 sub squares of the case $n=3$.
Then we have 3 cells left that we still have to fill don't we?
Can we align them so that we can put another L-square into it?
To do that we have to make the empty cell in that corner so that the three empty cells make a L, or not?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
To do that we have to make the empty cell in that corner so that the three empty cells make a L, or not?
Yes. So the sub squares at left-top, left-bottom, and right-bottom would have their empty cell at the center.
Those empty cells have the shape of an L then, allowing for another piece.

#### mathmari

##### Well-known member
MHB Site Helper
Yes. So the sub squares at left-top, left-bottom, and right-bottom would have their empty cell at the center.
Those empty cells have the shape of an L then, allowing for another piece.
I see!! Thanks a lot!!