This way the above pieces are 2x1-tetriamonds and 1x2-tetriamonds. Now we can colors the the six edge units and explore different sets:

- Sets of one-sided 2x1-parallelograms
- Sets of two-sided 2x1-parallelograms
- United sets of one-sided 2x1 and 1x2 parallelograms

Click the numbers of pieces to see some constructions.

Colors | Parallelogram Type | Number of Pieces | Total Number of Triangles |
Constructions |
---|---|---|---|---|

n | one-sided 2x1 | (n^6+n^3)/2 | (n^6+n^3)*2 | |

two-sided 2x1 | (n^6+n^3)/2 | (n^6+n^3)*2 | ||

one-sided 2x1 and 1x2 | (n^6+n^3) | (n^6+n^3)*4 | ||

2 | one-sided 2x1 | 36 | 144 | parallelograms |

two-sided 2x1 | 36 | 144 | parallelograms, hexagonal ring | |

one-sided 2x1 and 1x2 | 72 | 288 | parallelograms, rhomb, rings with triangle or hexagon symmetry | |

3 | one-sided 2x1 | 378 | 1512 | parallelograms, hexagonal ring |

two-sided 2x1 | 378 | 1512 | parallelograms | |

one-sided 2x1 and 1x2 | 756 | 3024 | parallelograms, hexagonal ring |

With this set you can make parallelograms of size 12x6 and 8x9 with uniformly red colored border.

The reflected parallelogram can't be constructed because the 9x8 parallelogram has a horizontal side of odd length and for the 6x12 parallelogram there are not enough pieces with red colored short edges. We have 16 pieces with one red colored short edge. But we need 24 edges for the border of the 6x12 parallelogram. If we place some of the 10 pieces with two red colored short edges at the border we need for each two pieces another one to balance the red colored short edges. Therefore we get no more than 6 additional unit edges and 16+6 < 24.

The same applies to the three parallelogram of size 6x24, 8x18, 9x16 and their reflections.

For constructions with triangle symmetry I divided the whole figure into three parts. For the left figure the size of the border parallel to the long sides is (12+8)*3=60 unit edges, but the 2-1 and 1-2 pieces can only contribute 30 and 26 unit edges, respectively. Therefore the hole got a green border.

Even a figure with hexagonal symmetry can be made.

Parallelograms of size 36x21, 28x27 and 12x63 are also possible but their reflections have odd horizontal lines, which are not suitable for 2x1 pieces.

The 84x9 parallelogram cannot be constructed. The long sides must be touched by 84 2x1 pieces with red colored long edges, but only 78 pieces are available.

For a hexagon of size16 with a hexagonal hole of size 2 I split the whole figure into 6 parts and preset the colors of the borders. It's important to balance the different color combinations at the borders.

3-fold constructions with three different colors at the border are possible, too. If you take a given piece and replace the colors 1,2,3 by 3,1,2 or 2,3,1, you get 126 triples of pieces. If you use only one piece of a triplet for a construction two other constructions are given by a color shift.. This way I made parallelogram of size 42x6, 21x12 and 6x42. Only the items with red borders are shown.

For the 24x63 parallelogram this method doesn't work, because the 12x63 parallelogram can only be made by 2x1 pieces but cannot be made by 1x2 pieces. Therefore an alternative partition was used. The following pictures are SVG-graphics to allow for high resolution prints.

Instead of searching partitions for the missing parallelograms I decided to look for a hexagonal ring of size 23 with a hole of size 7. The partition was rather difficult because you can't divide the ring into three congruent parts using lines form one outer corner to an inner one. After the inner ring of width 2 was finished I had to preset the border colors to balance the color combinations.

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