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Black and White Solids (Posted on 2006-03-04) Difficulty: 4 of 5

1)I have lots of black and white squares that can be joined together to make cubes. How many distinguishable cubes can I make?

2)Now I try it with triangles and regular octahedrons?

3)Now pentagons and regular dodecahedrons?

4)Triangles again but making regular icosahedrons!?

Note: Distinguishable means rotations are the same, but reflections are not.

No Solution Yet Submitted by Sir Percivale    
Rating: 4.0000 (1 votes)

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#3 - dodecahedron - Part 2 | Comment 5 of 11 |
(In reply to #3 - dodecahedron - Part 1 by Leming)

V.  The dodecahedron with five black faces might have:

  A.  All five touching each other
    1.  A black face to the table - four adjacent to the table.  (1)
    2.  A white face to the table - five adjacent to the table (1)
    3.  A white face to the table - four adjacent to the table
      a.  One adjacent to the black face on the left of the first row.  (This places it in the second row.)     (1)
      b.  One adjacent to the two black faces on the left of the first row (1)
      c.  One adjacent to the middle two black faces (1)
      d.  One adjacent to the right two black faces (1)
      e.  One adjacent to the right black face (1)
    4.  A black face to the table - three adjacent to the table,
      a.  The fifth in the second row adjacent either the left or right-most black face in the first row.  (1)
      b.  The fifth in the second row adjacent to two of the black faces in the first row.  (1)
    5.  A white face to the table - three black faces adjacent to the table,
      a.  The fourth and fifth black face in the second row - one adjacent to the left-most the other adjacent to the right-most black face.  (1) (from some points of view looks like a line) [Running total 31]

  B.  Four (of five) touching.
    1.  A black face to the table - three adjacent to the table - next to each other.  Fifth black face opposite the table.  (1)
    2.  A black face to the table - two adjacent to the table and to each other.  The fourth adjacent to the table but not the second and third.  The fifth opposite the table.  (1)
    3.  A white face to the table - four adjacent to the table - Fifth opposite the table.  (1)

  C.  Three (of five) touching
    1.  A black face to the table two adjacent to the table and to each other.  Two remaining face opposite one of the first three.  (1)
    2.  A white face to the table three black faces adjacent to the table and each other.  Fourth opposite the table.  Fifth opposite center of three black faces in the first row.  (1) [Running total  - 36]

VI.  The dodecahedron with six black faces might have:
  A.  All six touching each other
    1.  A black face to the table - five adjacent to the table.  (1)
    2.  A white face to the table - five adjacent to the table - the sixth touching two in the first row.  (1)
    3.  A black face to the table - four adjacent to the table
      a.  One adjacent to the black face on the left of the first row.  (This places it in the second row.)     (1)
      b.  One adjacent to the two black faces on the left of the first row (1)
      c.  One adjacent to the middle two black faces (1)
      d.  One adjacent to the right two black faces (1)
      e.  One adjacent to the right black face (1)
    4.  A white face to the table - four adjacent to the table
      a.  The fifth adjacent to the left most first row black face.  The sixth adjacent to the two black faces furthest to the left.  (1)
      b.  The fifth adjacent to the left most.  The sixth adjacent to the middle two.  (1)
      c.  The fifth adjacent to the left most.  The sixth adjacent to the right-most two.  (1)
      d.  The fifth adjacent to the left most.  The sixth adjacent to the right-most.  (1)
      e.  The fifth adjacent to the left-most two black faces on the first row.  The sixth adjacent to the right-most black faces of the first row.  (1)
      f.  The fifth adjacent to the left-most two black faces.  The sixth adjacent to the right-most.  (1)
      g.  The fifth adjacent to the middle two black faces.  The sixth adjacent to the right-most.  (1)
      h.  The fifth adjacent to the right-most two black faces.  The sixth adjacent to the right-most.  (1)
      i.  The fifth adjacent to the left-most black face.  The sixth opposite the table.  (1)
      j.  The fifth adjacent to the right-most black face.  The sixth opposite the table.  (1)
    5.  Three white faces sharing a vertex.  The opposite side also has three white faces sharing a vertex.  The remaining six are black. (1)

  B.  Five (of six) touching each other
    1.  A black face to the table - four adjacent to the table.  Sixth opposite the table.  (1)
    2.  A white face to the table - five adjacent to the table.  Sixth opposite the table.  (1)

  C.  Four (of six) touching each other
    1.  A black face to the table.  Three adjacent to the table and to each other.  One opposite the table and one opposite the center of the three in the first row.  (1)

  D.  Three (of six) touching each other
    1.  Three that share one vertex and the opposite three.  (1) [58 total]


  Posted by Leming on 2006-03-04 19:50:05
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