How Can You Tile an $8 \times 8$ Chessboard with L-Shaped Pieces?

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    2015
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SUMMARY

The problem involves tiling an $8 \times 8$ chessboard with one square removed using L-shaped pieces that each cover three squares. The challenge is to determine if it is possible to cover the remaining squares completely. The consensus from the discussion is that it is impossible to tile the chessboard under these conditions due to the parity of the squares and the nature of the L-shaped pieces. Specifically, the removal of one square disrupts the balance required for complete coverage.

PREREQUISITES
  • Understanding of chessboard parity and coloring principles.
  • Familiarity with combinatorial tiling problems.
  • Knowledge of L-shaped piece geometry and coverage.
  • Basic problem-solving skills in discrete mathematics.
NEXT STEPS
  • Research combinatorial tiling strategies in discrete mathematics.
  • Explore proofs related to parity in tiling problems.
  • Study variations of the L-shaped piece tiling problem.
  • Investigate other geometric shapes used in tiling puzzles.
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Mathematicians, puzzle enthusiasts, educators, and students interested in combinatorial geometry and tiling problems.

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Here is this week's POTW:

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Tile an $8 \times 8$ chessboard, with one square missing, with L-shaped pieces that cover three squares each. The missing square is arbitrary.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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No one solved this week's POTW. Here is my solution below:
You sort of recursively move outward from the missing square, as shown:

View attachment 4664

Note I haven't completely covered the $8\times 8$ square, but you should be able to see the pattern, and how you could complete it.
 

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