The Tetracairos belong to the family of Polycairos. They are composed of pentagons that include two right interior angles. These pentagons are suitable for tessellation, meaning they tile the plane without gaps or overlaps.

There are 2 Dicairos, 5 Tricairos, 17 Tetracairos, 55 Pentacairos, 206 Hexacairos, 781 Heptacairos, and 3,099 Octacairos.

Within the grid, every Tetracairo tile can be rotated by 90 degrees. In any such orientation, the pentagons that make up a tile point either horizontally or vertically.

For 12 of the 17 Tetracairos, the distribution of orientations is always balanced: half of the pentagons point horizontally and the other half vertically, regardless of rotation. The remaining 5 Tetracairos are asymmetric: three pentagons point in one direction and only one in the other (either 3:1 or 1:3).

When all 17 Tetracairos are assembled into a single shape, these imbalances add up, producing a difference of 2, 6, or 10 between the total number of horizontally and vertically oriented pentagons.

A mirror-symmetric shape, however, would require an even balance – that is, a difference of 0, 4, 8, and so on. Because this condition cannot be met with the full set of 17 Tetracairos, no mirror-symmetric shape made from all of them can exist.

The 3D-print templates consist of the 17 Tetracairo puzzle pieces and a box.

The Tetracairo Puzzle solver is part of a program collection available for download on Itch.io. It is for Windows only.

IMPORTANT: For a correct installation, please read the Instructions before unpacking!