Tetrad (geometry puzzle)

In geometry, a tetrad is a set of four simply connected disjoint planar regions in the plane, each pair sharing a finite portion of common boundary. It was named by Michael R. W. Buckley in 1975 in the Journal of Recreational Mathematics. A further question was proposed that became a puzzle, whether the 4 regions could be congruent, with or without holes, other enclosed regions.^[1]

Tetrad with one central region and 3 surrounding ones

Tetrad with a hole

Fewest sides and vertices

The solutions with four congruent tiles include some with five sides.^[2] However, their placement surrounds an uncovered hole in the plane. Among solutions without holes, the ones with the fewest possible sides are given by a hexagon identified by Scott Kim as a student at Stanford University.^[1] It is not known whether five-sided solutions without holes are possible.^[2]

Kim's solution has 16 vertices, while some of the pentagon solutions have as few as 11 vertices. It is not known whether fewer vertices are possible.^[2]

Congruent polyform solutions

Gardner offered a number of polyform (polyomino, polyiamond, and polyhex) solutions, with no holes.^[1]

• 
  11 squares
• 
  12 squares
• 
  10 triangles
• 
  22 triangles
• 
  26 triangles
• 
  4 hexagons