Ken's Puzzle of the Week
Tetromino Tiling
 

There are five basic tetrominos, as seen above.

1. Why can't you place all five into a 4x5 rectangle?  Reflections and rotations are allowed.
    
2. 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?
    
3. 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.
    

4. 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.

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.