1) Use parity / coloring argument to show that a 10x10 board cannot be tiled with 1x4 pieces.
2) Show that if a 7x7 board is tiled with 1x3 pieces, then the untiled square must occupy one of the following 9 positions: the center, one of the four corners, or one of the central squares along the sides of the board.
- A blog entry discusses the problem of tiling with dominoes with 4 squares removed.
- Edward Dijkstra has posted a short note on the following problem: Consider tiling a chessboard with dominoes. A domino whose squares are in the same row we call a horizontal domino. We claim that the number of horizontal dominoes with a black left square and the number of horizontal dominoes with a white left square are equal. [Corollary: the number of horizontal dominoes is even.]
- Free Chapter 1 (in PDF format) from "Across the Board: The Mathematics of Chessboard Problems" by John J. Watkins, 2004 — the chapter discusses several covering problems using dominoes and chess pieces.
- Tilings by Federico Ardila and Richard P Stanley, Clay Public Lecture, 2004.
- The Inquisitive Problem Solver (MAA Problem Book Series) (2002, 344 pages) by Paul Vaderlind, Richard K. Guy & Loren C. Larson discusses the general problem of rook tours with multiple forbidden squares
Solutions for Followup Problems listed above:
1) Label rows as 0, 1, ..., 9. Label columns similarly. A square in column c and row r is assigned the color (c + r) mod 4. The sum of colors covered by any 1x4 tile is 2 modulo 4. The sum of colors of 25 such tiles shall also be 2. However, the sum of colors of all 100 squares is 0.
2) Color the seven rows as 0, 1, 2, 0, 1, 2, 0. The sum of numbers covered by a tile is always divisible by 3. So the uncovered square must belong to a row labeled 0. Now repeat the same argument with 'rows' replaced by 'columns'. The intersection of these rows and columns are the 9 squares listed in the problem statement.