Tiling rectangles and half strips with congruent polyominoes
A polyomino is rectifiable if it tiles a rectangle. The rectangular order of a rectifiable polyomino is the smallest number of copies of it which form a rectangle. A rectifiable polyomino is odd if there is some rectangle tiled by an odd number of copies of it; otherwise it is even. The odd order of an odd polyomino is the smallest odd number of copies of that polyomino that form a rectangle. Using a computer program which searches for shortest cycles in an appropriately constructed graph, the author has found three new rectifiable polyominoes. Although the rectangles formed are the smallest found to date, it is not known whether they are minimal. \textit{S. W. Golomb} [``Tiling rectangles with polyominoes'', Math. Intell. 18, 38-47 (1996)] has exhibited several rectifiable ``boot'' polyominoes, and has raised the question of whether other boot polyominoes have this property. A partial solution is given here by showing that the boot polyomino of size \(8n+2\) tiles a \((24n+10) \times (112n+28)\) rectangle, and thus has rectangular order \(\leq 336n+140\). The author also displays several infinite families of odd polyominoes, and is able to establish that there is no upper bound to the odd order of polyominoes. He does this by considering the family of so-called \(L\) polyominoes, proving that if \(n\) is prime, then the \((n+2) \times 3n\) rectangle is the unique minimal odd rectangle for an \(L\) polyomino of size \(n\). A conjecture is that the condition that \(n\) be prime is unnecessary. The section on odd polyominoes is concluded by the display of an even polyomino. Another open question is whether every polyomino that tiles a half strip also tiles a rectangle. The paper gives three new infinite families of polyominoes that tile half strips. While some members of these families are known to tile rectangles, none are known not to tile rectangles. The paper concludes with a set of remaining open problems.
scientific article; zbMATH DE number 1076994