T-Tetrominos in Arithmetic Progression

Abstract

A famous result of D. Walkup is that an $m\times n$ rectangle may be tiled by T-tetrominos if and only if both $m$ and $n$ are multiples of 4. The "if" portion may be proved by tiling a $4\times 4$ block, and then copying that block to fill the rectangle; but, this leads to regular, periodic tilings. In this paper we investigate how much "order" must be present in every tiling of a rectangle by T-tetrominos, where we measure order by length of arithmetic progressions of tiles.

Figures (11)

• Figure 1: (Left) A 4x4 unit. (Middle) A periodic tiling of a $20\times 20$ square. (Right) A tiling without any 3-term AP of tiles.
• Figure 2: Boundary tiling with no arithmetic progression of length 2.
• Figure 3: Tilings of $4\times l$ rectangles are made of units like these.
• Figure 4: Names for the orientations and vertical locations of all T-tetrominos in a $4\times N$ tiling.
• Figure 5: A tiling (a), its $2\times 2$ blocks (b), its chain graph (c), and the correspondence between tiles and shaded arrows (d). The two shaded tiles are discussed in the proof of Lemma \ref{['lemma-bijection']}.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3: van der Waerden's Theorem
• Corollary 4
• Lemma 5: mod 4 Lemma
• Lemma 6: d1 Lemma
• Lemma 7: AB Lemma
• Lemma 8
• Lemma 9
• Lemma 10: The dx/dy Lemma