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Can we tile the board by L trominos? [closed]

Counting board states in a game?$ntimes n$ board, non-challenging rooksnumber of ways in which the board can be restored to a winning configurationFilling an $ntimes n$ boardThree knights on a 3x3 chess boardFind the total number of ways in which the board can be shaded in a particular way.Rooks on an incomplete chessboardNumber of ways a board can be found within a gameCovering a board problem

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4

$begingroup$

I tried to cover this, but there is no way I can fill it.
The black square is a removed square.

Is there a way to prove that it cannot be filled with L trominos?

combinatorics

share|cite|improve this question

edited Aug 10 at 21:21

Tanner Swett

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asked Aug 10 at 12:15

Angelo MarkAngelo Mark

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$endgroup$

closed as off-topic by Peter, Lee David Chung Lin, Xander Henderson, Don Thousand, The Count Aug 11 at 0:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Peter, Lee David Chung Lin, Xander Henderson, Don Thousand, The Count

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  L shaped using three squares= L tromino
  $endgroup$
  – Angelo Mark
  Aug 10 at 12:22
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  There are only $24$ squares.
  $endgroup$
  – Culver Kwan
  Aug 10 at 12:23
  
  

  
  
  
  

add a comment | 

4

$begingroup$

I tried to cover this, but there is no way I can fill it.
The black square is a removed square.

Is there a way to prove that it cannot be filled with L trominos?

combinatorics

share|cite|improve this question

edited Aug 10 at 21:21

Tanner Swett

6,2742222 silver badges4242 bronze badges

asked Aug 10 at 12:15

Angelo MarkAngelo Mark

4,27322 gold badges1717 silver badges4343 bronze badges

$endgroup$

closed as off-topic by Peter, Lee David Chung Lin, Xander Henderson, Don Thousand, The Count Aug 11 at 0:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Peter, Lee David Chung Lin, Xander Henderson, Don Thousand, The Count

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  L shaped using three squares= L tromino
  $endgroup$
  – Angelo Mark
  Aug 10 at 12:22
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  There are only $24$ squares.
  $endgroup$
  – Culver Kwan
  Aug 10 at 12:23
  
  

  
  
  
  

add a comment | 

$begingroup$

I tried to cover this, but there is no way I can fill it.
The black square is a removed square.

Is there a way to prove that it cannot be filled with L trominos?

combinatorics

share|cite|improve this question

edited Aug 10 at 21:21

Tanner Swett

6,2742222 silver badges4242 bronze badges

asked Aug 10 at 12:15

Angelo MarkAngelo Mark

4,27322 gold badges1717 silver badges4343 bronze badges

$endgroup$

share|cite|improve this question

edited Aug 10 at 21:21

Tanner Swett

6,2742222 silver badges4242 bronze badges

asked Aug 10 at 12:15

Angelo MarkAngelo Mark

4,27322 gold badges1717 silver badges4343 bronze badges

share|cite|improve this question

edited Aug 10 at 21:21

Tanner Swett

6,2742222 silver badges4242 bronze badges

asked Aug 10 at 12:15

Angelo MarkAngelo Mark

4,27322 gold badges1717 silver badges4343 bronze badges

edited Aug 10 at 21:21

Tanner Swett

6,2742222 silver badges4242 bronze badges

edited Aug 10 at 21:21

Tanner Swett

6,2742222 silver badges4242 bronze badges

asked Aug 10 at 12:15

Angelo MarkAngelo Mark

4,27322 gold badges1717 silver badges4343 bronze badges

asked Aug 10 at 12:15

Angelo MarkAngelo Mark

4,27322 gold badges1717 silver badges4343 bronze badges

1 Answer
1

active

oldest

votes

11

$begingroup$

We color each square at $i$-th row and $j$-row with both $i,j$ odd with red paint. Now the $9$ red squares needs $9$ L trominos to be covered. On the other hand, to cover the $5^2-1=24$ squares we need $24/3=8$ L trominos. Contradiction!

share|cite|improve this answer

edited Aug 10 at 14:19

answered Aug 10 at 12:26

Robert ZRobert Z

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$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Your answer is very good! How can you think of this?
  $endgroup$
  – Culver Kwan
  Aug 10 at 12:27
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  (+1) I had already solved this problem moths ago, but by considering lots of possible cases. This is a great approach.
  $endgroup$
  – José Carlos Santos
  Aug 10 at 12:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Wow, thanks :) nice +1
  $endgroup$
  – Angelo Mark
  Aug 10 at 12:33
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @CulverKwan Coloring in Tiling problem is quite common: see for example pregatirematematicaolimpiadejuniori.files.wordpress.com/2016/07/…
  $endgroup$
  – Robert Z
  Aug 10 at 12:44
  

  
  
  
  

add a comment | 

11

$begingroup$

We color each square at $i$-th row and $j$-row with both $i,j$ odd with red paint. Now the $9$ red squares needs $9$ L trominos to be covered. On the other hand, to cover the $5^2-1=24$ squares we need $24/3=8$ L trominos. Contradiction!

share|cite|improve this answer

edited Aug 10 at 14:19

answered Aug 10 at 12:26

Robert ZRobert Z

110k1010 gold badges7878 silver badges153153 bronze badges

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Your answer is very good! How can you think of this?
  $endgroup$
  – Culver Kwan
  Aug 10 at 12:27
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  (+1) I had already solved this problem moths ago, but by considering lots of possible cases. This is a great approach.
  $endgroup$
  – José Carlos Santos
  Aug 10 at 12:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Wow, thanks :) nice +1
  $endgroup$
  – Angelo Mark
  Aug 10 at 12:33
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @CulverKwan Coloring in Tiling problem is quite common: see for example pregatirematematicaolimpiadejuniori.files.wordpress.com/2016/07/…
  $endgroup$
  – Robert Z
  Aug 10 at 12:44
  

  
  
  
  

add a comment | 

11

$begingroup$

We color each square at $i$-th row and $j$-row with both $i,j$ odd with red paint. Now the $9$ red squares needs $9$ L trominos to be covered. On the other hand, to cover the $5^2-1=24$ squares we need $24/3=8$ L trominos. Contradiction!

share|cite|improve this answer

edited Aug 10 at 14:19

answered Aug 10 at 12:26

Robert ZRobert Z

110k1010 gold badges7878 silver badges153153 bronze badges

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Your answer is very good! How can you think of this?
  $endgroup$
  – Culver Kwan
  Aug 10 at 12:27
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  (+1) I had already solved this problem moths ago, but by considering lots of possible cases. This is a great approach.
  $endgroup$
  – José Carlos Santos
  Aug 10 at 12:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Wow, thanks :) nice +1
  $endgroup$
  – Angelo Mark
  Aug 10 at 12:33
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @CulverKwan Coloring in Tiling problem is quite common: see for example pregatirematematicaolimpiadejuniori.files.wordpress.com/2016/07/…
  $endgroup$
  – Robert Z
  Aug 10 at 12:44
  

  
  
  
  

add a comment | 

$begingroup$

We color each square at $i$-th row and $j$-row with both $i,j$ odd with red paint. Now the $9$ red squares needs $9$ L trominos to be covered. On the other hand, to cover the $5^2-1=24$ squares we need $24/3=8$ L trominos. Contradiction!

share|cite|improve this answer

edited Aug 10 at 14:19

answered Aug 10 at 12:26

Robert ZRobert Z

110k1010 gold badges7878 silver badges153153 bronze badges

$endgroup$

share|cite|improve this answer

edited Aug 10 at 14:19

answered Aug 10 at 12:26

Robert ZRobert Z

110k1010 gold badges7878 silver badges153153 bronze badges

answered Aug 10 at 12:26

Robert ZRobert Z

110k1010 gold badges7878 silver badges153153 bronze badges

answered Aug 10 at 12:26

Robert ZRobert Z

110k1010 gold badges7878 silver badges153153 bronze badges