Polyiamonds with 1..6 Triangles

We have 1+1+1+3+4+12=22 two-sided pieces and 1+1+1+4+4+19=30 one-sided pieces.

Three Sets of Two-sided Polyiamonds with 1 up to 6 Triangles

There are 66 pieces with a total of 330 triangles. Since 330=6*(8²-3²) we can get a regular hexagon with side length 8 and a hexagonal hole of side length 3.

Four Sets of of Two-sided Polyiamonds with 1 up to 6 Triangles

There are 88 pieces with a total of 440 triangles. A hexagon and a figure with two axis of symmetry are already shown in the title, here is another one with the real pieces.

Since 440=2*220 and 220 = 20*11 = 22*10 = 44*5 = 55*4 you can try to solve four parallelograms.

Solving 440 = a² - b² leads to three trapezia.

If 440 is the sum of three square numbers, three triangles can be constructed. One example is here.

Three Sets of One-sided Polyiamonds with 1 up to 6 Triangles

There are 90 pieces with a total of 498 triangles. Since 498=6*74 we can try to combine 6 congruent parts with 74 triangles to get a figure with rotational symmetry.

Four Sets of One-sided Polyiamonds with 1 up to 6 Triangles

There are 120 pieces with a total of 664 triangles. Since 664=2*(2*166)=2*(2*2*83) a parallelogram of size 83x4 can be made.

With 664= 18² + 18² + 4² we can also construct three triangles of size 18, 18 and 4.

Polyiamonds with 4..6 Triangles

We have 3+4+12=19 two-sided pieces and 4+4+19=27 one-sided pieces.

Three Sets of Two-sided Polyiamonds with 4 up to 6 Triangles

There are 57 pieces with a total of 312 triangles. This allows for constructions with rotational symmetry.

Four Sets of Two-sided Polyiamonds with 4 up to 6 Triangles

There are 76 pieces with a total of 416 triangles. Hexagons of size 1-6-1-1-6-1 need 26 triangles. I stacked 16 of them to get a figure with two axis of symmetry and 26*16=416 triangles. You can see 4 congruent parts only different in color.

Three Sets of One-sided Polyiamonds with 4 up to 6 Triangles

There are 81 pieces with a total of 480 triangles, just enough for a regular hexagon of size 9 with a hexagonal hole of size 1.

Four Sets of One-sided Polyiamonds with 4 up to 6 Triangles

There are 108 pieces with a total of 640 triangles. I had a look at multiple constructions like eight parallelograms or five rhombs. If we check for matching borders, we can combine four parallelograms of size 20x4 to get 80x4 or 16x20 parallelograms.

Polyiamonds with 5 or 6 Triangles

We have 4+12=16 two-sided pieces and 4+19=23 one-sided pieces.

Three Sets of Two-sided Polyiamonds with 5 or 6 Triangles

There are 48 pieces with a total of 276 triangles. A ring with hexagonal symmetry is shown.

Four Sets of Two-sided Polyiamonds with 5 or 6 Triangles

There are 64 pieces with a total of 368 triangles. Since 368 mod 3 = 2, no figures with rotational symmetry are possible, but we have a convex hexagon with two axis of symmetry.

Three Sets of One-sided Polyiamonds with 5 or 6 Triangles

There are 69 pieces with a total of 432 triangles. Since 432=6*(9² - 3²) I looked for another hexagoanl ring.

Four Sets of One-sided Polyiamonds with 5 or 6 Triangles

There are 92 pieces with a total of 576 triangles just enough for a hexagonal ring, because 576= 6*(10² - 2²).

Polyiamonds with 6 or 7 Triangles

We have 12+24=36 two-sided pieces and 19+43=62 one-sided pieces.

Three Sets of Two-sided Polyiamonds with 6 or 7 Triangles

There are 108 pieces with a total of 720 triangles. We have 720=6*(11² - 1²) for a hexagonal ring and 720 = 3* (6*6² + 6*2²) for multiple constructions of regular hexagons.

Four Sets of Two-sided Polyiamonds with 6 or 7 Triangles

There are 144 pieces with a total of 960 triangles. 960 = 6*(13²-3²) allows a hexagonal ring.

Three Sets of One-sided Polyiamonds with 6 or 7 Triangles

There are 186 pieces with a total of 1245 triangles. Since hexagonal symmetry isn't possible, we can look for integer solutions of 1245=a²-2b² -c² to get hexagons with one axis of symmetry.

Four Sets of One-sided Polyiamonds with 6 or 7 Triangles

There are 248 pieces with a total of 1660 triangles. Take a triangle of size 36 and cut off one triangle of size 16 and two triangles of size 10. This way you get a hexagon with one axis of symmetry.

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