Polyform Counts

Units are unit line segments on the square grid.

The polysticks (fully-connected):

"*" above means that pieces with enclosed holes exist.

The quasi-polysticks (includes disconnected forms that have gaps of maximum length 1):

Shapes

Holes (denoted by a "*" in the function name) consist of internal segments only, no circumference segments.

Grid size definitions are OK (instead of side length), as long as it's clearly noted.

Rectangular MxN grid (bordered):

R(m,n) = m * (n - 1) + n * (m - 1)

Rectangular MxN grid (unbordered):

R*(m,n) = (m - 2) * (n - 1) + (m - 1) * (n - 2)

MxN diamond lattice:

D(m,n) = m * n * 4

Triangles (n == height):

T(n) = n(n + 3)

Double Triangles (n == height):

DT(n) = 2n² +3n - 1

Aztec Diamonds (n == side length or height of quadrant; _A134582):

A(n) = 4(n + 1)² - 4
     = 4n(n + 2)

Trapezoid (m = base / jagged hypotenuse, n=smooth side length):

Tr(m,n) = T(m) - T(m - n) + 2(m - n)

Parallelogram (m=smooth base length, n=jagged side length):

P(m,n) = R(m,n) + 2(n - 1)

Parity imbalance = (n - 1).

Potential Puzzles

Puzzles not otherwise noted below have not been implemented or solved.

Initial numbers are the counts of unit line segments in the puzzles.

15: tristicks

20: polysticks of order 1 to 3

21: one-sided tristicks

26: one-sided polysticks of order 1 to 3

• DT(3)
• T(4) - 2 segments (= trapezoid)

40: one-sided welded tetrasticks

• No solutions:
  • R*(6,6)
  • D(5,2)

49: seven-segment digits

• No solutions:
  • R*(7,6)

60: 15/16 tetrasticks

• No solutions:
  • D(5,3) (= Tetrasticks3x5DiamondLattice)
  • R*(7,7)

64: tetrasticks (must have parity imbalance = 2, 6, 10, or 14)

• Potential:
• No solutions:
  • D(4,4) diamond lattice -- impossible due to parity (imbalance = 0)
  • P(5,5) -- impossible due to parity (imbalance = 4; = TetrasticksParallelogram5x5)
  • DT(5)

84: polysticks of order 1 to 4

• Potential:
  • T(8) - 4 segments
  • T(9) - R*(5,5)
  • R*(8,8) (probably no solutions; = Polysticks1234_8x8Unbordered)
• No solutions:
  • D(5,5) - D(2,2) (5x5 diamond lattice ring; = Polysticks1234_5x5DiamondLatticeRing)

100: one-sided tetrasticks

• P(5,8) (impossible due to odd parity imbalance?)

126: one-sided polysticks of order 1 to 4

• R(12,6)
• R*(13,7)
• R(9,8) - 1 segment
• T(10) - 4 segments

Misc

• Tetrasticks horizontal/vertical parity imbalance (abs(vertical - horizontal)):

  I L Y V T X U N J H F Z R W P O
  4 2 2 0 0 0 0 2 2 2 0 0 0 0 0 0
  

  Total minimum imbalance = 2, therefore for symmetrical puzzles one of HJLNY must be omitted.

  For one-sided tetrasticks, only even imbalances are possible.

Links

• The "Nexos" board game, designed by the designer of the "Blokus" games, uses the 24 polysticks of order 1-4.