##
Ken's Puzzle of the Week

Tetromino Tiling

There are five basic tetrominos, as seen above.

- Why can't you place all five into a 4x5 rectangle? Reflections and
rotations are allowed.

- Place all five into a 3x7 rectangle. If you color the grid as a
checkerboard, with the corners black, which color is uncovered? Can
you find a different configuration which leaves the other color uncovered?

- If we color the pieces in a checkerboard pattern, we have seven
unique pieces.

Can these seven pieces be placed into a 4x7 rectangle, to create a
checkerboard pattern? Reflections and rotations
are allowed.

- If we color them and don't allow reflections, we have 10 unique pieces.

Can these 10 pieces be placed into a 5x8 rectangle to create a checkerboard
pattern? Into a 4x10 rectangle? Only rotations are allowed.

Source: Original.

Solutions were received from Kirk Bresniker.1. No. Consider
coloring the grid and pieces as a checkerboard. The T-piece must cover
three of one color and one of another. All other pieces cover 2 of
each color, so the pieces cover 11 of one color and 9 of the other.
The grid has 10 of each color, so cannot be covered.

2. With the corners black, there are 11 black and 10 white. The
T-piece forces us to cover 11 of one color and 9 of the other. A white
square will always be left uncovered.

3 & 4. See Kirk's powerpoint slides.
Each of the puzzles has a solution.

Mail to Ken