Conditions for Constructions
You can imagine a lot of constructions using the stacked tetrahexes, but before you try to find a solution with
these pieces you should check some conditions:
- The number of hexagons must be smaller or equal 92, because 23 pieces can only contribute 23*4=92 hexagons.
- The number of hexagons must be a multiple of 4, because all pieces are made of 4 hexagons.
- For constructions with height 1 only the 7 flat pieces with a total of 28 hexagons can be used.
- For constructions with height 2 only 19 pieces are available, because shoe, mug, column and skew bloc
have a height greater than 2. ( For the names see Set of Stacked Tetrahexes )
- If the height of a construction is 3 the column can't be used and the maximum number of hexagons is 22*4=88.
- If all pieces are used, the sum of hexagons in all odd layers (1, 3, 5,...) must be odd.
There are 7 pieces (shoe, 3 seals, 3 pedestals), which contribute an odd number of hexagons to both
the even and the odd layers of a construction. Therefore the sum must also be odd for both the odd layers and the even layers.
- For prisms you can't use pieces, which don't fit the cross-section.
Let's have a look at some impossible constructions.
- If the cross-section of a prism is a triangle of size 3, the height must be lower than 14.
Four pieces don't fit the cross-section and the left 19 pieces provide only 19*4=76 hexagons whereas 14*6=84 hexagons are neede for the prism
of height 14.
- You can't construct a prism of height 14 and a hexagonal cross-section made from seven hexagons.
Two pieces cannot be used because they don't fit the cross-section. The remaining set has an odd number of odd pieces,
contributing odd numbers of hexagons to odd layers but
the total number of hexagons in odd layers is even.
- A flat triangle of size 7 has an area of 28 hexagons and 7 flat pieces are available. This seems to be pretty enough.
But it was shown decades ago that this problem
isn't solvable analyzing the different positions for the propeller.