Polyform Counts

Units are unit line segments on the hexagonal grid.

The polytwigs (fully-connected):

The quasi-polytwigs (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.

Hexagons & holes:

H(n) = 9n² - 3n
     = 3n(3n - 1)

H*(n) = H(n) - 12n + 6
      = 9n² - 15n + 6

Triangles & holes:

T(n) = 3n/2(n + 3)
     = 3/2(n² + 3n)

T*(n) = T(n) - 6n
      = 3n/2(n - 1)
      = 3/2(n² - n)

Hexagrams & holes:

Hg(n) = H(n) + 6T(n - 1) - 12(n - 1)
      = 18n² - 6n - 6

Hg*(n) = Hg(n) - 24n + 18
       = 18n² - 30n + 12

Parallelograms (& staggered rectangles) & holes:

P(m,n) = 3mn + 2(m + n) - 1

Rs(m,n) = P(m,n)

P*(m,n) = P(m,n) - 4(m + n) + 2
        = 3mn - 2(m + n) + 1

Inset rectangles:

Ri(m,n) = P(m,n) - 3 * int(m / 2) - 1 + (m % 2)

Rounded inset rectangles (inset rectangles that start with the short column; identical [but flipped] for even m):

Rr(m,n) = Ri(m,n) - (5 * (m % 2))

Trapezoids & holes:

Tr(m,n) = m(3n + 2) - n(3n - 5)/2

Tr*(m,n) = Tr(m,n) - 2m - 4n - 2(m - n)
         = m(3n - 2) - n(3n - 1)/2

Elongated hexagons & holes:

He(m,n) = H(n) + (m - n)(6n - 1)
        = 3n(2m + n) - (m + 2n)

He*(m,n) = He(m,n) - 4m - 8n + 6
         = 3n(2m + n) - 5(m + 2n) + 6

Semi-regular hexagons & holes (two different side lengths, alternating; e.g. truncated triangles; m > n, n > 1):

Hs(m,n) = T(m + 2(n - 1)) - 3T(n - 1) + 6(n - 1)
        = 3/2(m² + n² + 4mn - m - n)

Hs*(m,n) = Hs(m,n) - 6m - 6n + 6
         = 3/2(m² + n² + 4mn - 5m - 5n + 4)

Chevrons & holes:

C(m,n) = P(m, 2n - 1)
       = 6mn - m + 4n - 3

C*(m,n) = C(m,n) - 4m - 8n + 6
        = 6mn - 5m - 4n + 3

Butterflies & holes:

B(m,n) = He(m,n) - 2P(n-1,n-1) + 8(n - 1) - 2
       = 6mn - 3n² - m + 10n - 6

B*(m,n) = B(m,n) - 4m - 8n + 6
        = 6mn - 3n² - 5m + 2n

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.

16: tetratwigs (polytwigs order 4)

• No solutions:
  • P(3,1)

24: one-sided tetratwigs (polytwigs order 4)

• No solutions:
  • Ri(3,2) = B(2,2)
  • T(3) - 3 segs (= OneSidedTetratwigsTriangle_5)
  • P(3,2) - 3 segs (= OneSidedTetratwigs3x2_1, OneSidedTetratwigs3x2_2)
  • Rs(3,2) - 3 segs (= OneSidedTetratwigsArch_5)

28: polytwigs order 1-4

39: one-sided polytwigs order 1-4

51: quasi-tritwigs

• 3 odd shapes, (email from Colin Brown, 2012-02-15)
• He(4,2) - 1 unit
• Tr(5,4) - 5 units
• Tr(7,2) - 4 units
• Ri(5,3) - 3 units
• Ri(4,4) - 5 units
• Rs(6,2)
• Rs(10,1)

58: quasi-polytwigs order 1-3

• T(5) - 2 units
• Tr*(10,3)
• Tr*(8,4)
• P(7,2) - 1 unit
• C(2,4) - 1 unit
• C(3,3) - 2 units
• C(5,2) - 2 units
• Ri(6,3) - 3 units

60: pentatwigs (polytwigs order 5)

• Potential:

  • Grid saturated with the 12 pentatwigs in a non-touching configuration. "Saturated" means that all vertices are occupied with no vacancies, and since the pentatwigs are non-touching, each vertex is occupied by exactly one polyform.

    Colin Brown provided an example, "rosette clustered trefoil". This configuration of pentatwigs exactly corresponds to the 12 hexiamonds. The pentatwig line segments join the centers of the hexiamonds' unit triangles, and the unoccupied hexagonal-grid line segments correspond to the borders between hexiamonds.

    The pentatwig nodes (end points of the line segments) can also be transformed into hexagons, resulting in a subset of the hexahexes (120° neighbors only) on a hexagonal tiling with regular sparse holes (1/3 of all hexagons are holes). (This is equivalent to the hexiamonds with the tips of each unit triangle removed.)

    See email from Colin F. Brown, 2012-04-29 & 2012-05-06.

    Implementation idea: intersection points are primary columns, line segments are secondary. That's just another way of specifying polyiamonds though, exactly equivalent.

  • H(2) & T(3) + 3 spikes ["A simultaneous construction" in email from Colin F. Brown, 2012-05-15].

• No solutions:

  • H(3) - H*(2) (= PentatwigsHexagonRing)
  • 2H(2) (= PentatwigsTwoHexagons)
  • B(4,3) - 5 units (= PentatwigsButterfly_X)
  • H(3) - 12 units (3 corners & 3 internal units; = PentatwigsTrefoil_x1)
  • H(2) + 5H(1) on corners (= PentatwigsSnowflake)
  • PentatwigsTrefoilRing_x
  • Hs(4,2) - T*(3) (= PentatwigsSemiregularHexagonRing_x)
  • PentatwigsTrefoil_x2
  • He(5,2) - P*(4,1) (= PentatwigsElongatedHexagonRing_x)
  • Hg(2) + 6 segments (= PentatwigsHexagram_x)

76: tetratwigs & pentatwigs (polytwigs order 4 & 5)

• P(15,1)
• Ri(11,2)
• H(3) + 4 units
• T(5) + 16 units
• Tr(6,5) - 1 unit
• He(2,4) - 10 units
• Hs(4,3) - Hs*(3,2) + 1 unit
• Hs(5,2) - 17 units
• B(4,4) - 2 units
• B(6,3) - 23 units
• more solutions to four congruent groups?

84: one-sided quasi-tritwigs

88: polytwigs order 1-5

93: one-sided quasi-polytwigs order 1-3

95: one-sided pentatwigs (polytwigs order 5)

• more designs from Peter F. Esser (see email 2012-06-03)

119: one-sided polytwigs order 4 & 5 (tetratwigs & pentatwigs)

134: one-sided polytwigs order 1-5

• Ri(14,3) - 3 segs
• Rr(7,6) - 3 segs
• C(12,2) - 3 segs

162: hexatwigs (polytwigs order 6)

• P(9,5)
• Rs(9,5)
• Tr(11,5)
• C(9,3)
• C(5,5)

250: polytwigs order 1-6

• P(12,6) - 1 seg
• B(15,2) - 2 segs
• Ri(11,7) - 1 seg
• Ri(13,6) - 3 segs

294: one-sided hexatwigs (polytwigs order 6)

• Hs(8,4)
• Ri(13,7)

428: one-sided polytwigs order 1-6

• Ri(15,9) - 3 segs

546: heptatwigs

• H(8) - 6 segments (e.g. set(((1,7,1), (1,14,2), (7,1,0), (7,14,0), (14,1,2), (14,7,1))) )
• He(11,10) - He*(8,7) - 4 segs (design by Peter F. Esser, email 2012-02-21)
• 3(He(16,2) - 2), separate strips (design by Peter F. Esser, email 2012-05-05)

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