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

References
  1. Martin Gardner, Penrose Tiles to Trapdoor ciphers, 1989, p.121-123
  2. Further Questions about Tetrads by Walter Trump
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