Tiling Square with Tetrominoes Fault-Free

There are five types of tetromino (a polyomino that consist of four squares):

These are called "straight", "square", "L-tetromino", "T-tetromino", and "skew" or "Z-tetromino".

Four types of tetromino (all, except the skew) tile an 8×8 chessboard. In fact, each of these already tiles a 4×4 square:

The covering of an 8×8 chessboard in this manner with a unique type of tetromino would introduce fault lines - the straight lines between squares that run from edge to edge.

The first two claims are almost obvious or become so after short experimentation. The simplest way is to start from a corner tile and see how a tiling may evolve from there. (This is also a good approach to proving that the skew tromino does not tile a rectangle, let alone a square.)

There is a very general result (1981) by Ron Graham for the existence tiling rectangular boards with rectangular pieces:

A solution to the third one is shown below.

George Jelliss found 12 solutions to the fourth problem.

References

1. R. L. Graham, Fault-Free Tilings of Rectangles, In The Mathematical Gardner: A Collection in Honor of Martin Gardner (Ed. D. A. Klarner), Van Nostrand Reinhold (March 1982)

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One solution is found at [Martin, p. 48]:

References

1. G. E. Martin, POLYOMINOES: A Guide to Puzzles and Problems in Tiling, MAA, 1996

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