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

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,008
Hey!! :eek:

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:

L_stones.png

Is the drawing correct?


(Wondering)
 
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mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,008
Can we use the sketch of the case $n=2$ to get the one of the case $n=3$ ? (Wondering)

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.

(Wondering)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,687
Can we use the sketch of the case $n=2$ to get the one of the case $n=3$ ? (Wondering)

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? (Wondering)
 

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,008
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? (Wondering)
To do that we have to make the empty cell in that corner so that the three empty cells make a L, or not? (Wondering)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,687
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. (Thinking)
 

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,008
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. (Thinking)
I see!! Thanks a lot!! (Mmm)