During our holidays in summer 2002 I saw this ancient Roman mosaic, which tiles the plane with squares
whose straight sides are replaced by arcs. This is a basic cell to make polyforms a little bit different from
the usual polyominoes. Putting four or five basic cells together you get 7 or 21 distinct pieces respectively. With the seven pieces you can easily pack a 7x4 rectangle but to fill a 7x15 rectangle with the 21 penta-pieces will take you some time. Miro Vicher has this puzzle on his site, too. For larger constructions a computer program would be helpful. |

The following table shows the number of pieces with n basic cells. Depending on whether the pieces can be flipped (two-sided) or not (one-sided), we get different counts. Subsets are given if the pieces are made up from arc-dominoes. Click on the numbers to see some constructions with the sets.

Number of Squares | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Two-sided | 1 | 1 | 3 | 7 | 21 | 60 | 208 | 704 | |

Made up from Dominoes | - | 1 | - | 5 | - | 39 | - | 394 | |

One-sided | 1 | 1 | 4 | 11 | 35 | 110 | 392 | 1372 | |

Made up from Dominoes | - | 1 | - | 9 | - | 72 | - | 777 |

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5 made up from dominoes, total area = 20.

11 one-sided pieces, total area = 44.

9 made up from dominoes, total area = 36.

Using the last set you can ask for 3-fold replicas ot the tetrominoes. Unfortunately a construction for the L and skew-tetromino failed. For the T-tetromino it's impossible due to a checkering argument.

Three

You can see the