Tiling a Square with T-, L-, and a Square Tetrominoes

Both L- and T-tetrominoes tile a chessboard. It takes 16 tiles of each kind. The question I wish to ask is whether

  1. Is it possible to cover a chessboard with 15 T-tetrominoes and a square one?
  2. Is it possible to cover a chessboard with 15 L-tetrominoes and a square one?

It is known that a straight tromino (or even three of them) could not be replaced with square tiles. What is happening with T- and L-tetromino tilings?

Solution

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Solutions

  1. Is it possible to cover a chessboard with 15 T-tetrominoes and a square one?

The answer is, No.

Impossibility of tiling of a chessboard with 15 T- and 1 square tetromino

In the common coloring of a chessboard, a T-tetromino covers an odd number of white (1 or 3) and an odd number (also, 1 or 3) of dark squares. It follows that an odd number (15 in particular) will cover an odd number of white and an odd number of dark squares. A square tile would add 2 white and 2 dark squares to such a tiling, leaving the totals odd. But the number of white and dark squares on the board is even, implying that no such tiling is possible.

  1. Is it possible to cover a chessboard with 15 L-tetrominoes and a square one?

The answer is again, No. Now we color the board differently:

Impossibility of tiling of a chessboard with 15 L- and 1 square tetromino

The argument is exactly the same. In the new coloring, an L-tetromino covers an odd number of cells of the same color, whereas a square tetromino always covers 2 cells of each color.

References

  1. Anany and Maria Levitin, Algorithmic Puzzles (Oxford University Press, 2011), #33(f)

Related material
Read more...

  • Covering A Chessboard With Domino
  • Dominoes on a Chessboard
  • Tiling a Chessboard with Dominoes
  • Vertical and Horizontal Dominoes on a Chessboard
  • Straight Tromino on a Chessboard
  • Golomb's inductive proof of a tromino theorem
  • Tromino Puzzle: Interactive Illustration of Golomb's Theorem
  • Tromino as a Rep-tile
  • Tiling Rectangles with L-Trominoes
  • Squares and Straight Tetrominoes
  • Covering a Chessboard with a Hole with L-Trominoes
  • Tromino Puzzle: Deficient Squares
  • Tiling a Square with Tetrominoes Fault-Free
  • Tiling a Rectangle with L-tetrominoes
  • Tiling a 12x12 Square with Straight Trominoes
  • Bicubal Domino
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