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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 1 | Comment 4 of 11 |

Edited:  I did not multiply my answer by two to account for all the images that will have a "negative" by swapping colors.

Thus I arrived at 116 total. I found 58 possibilities below (and in post #2). 

(0) The dodecahedron may be all white (1).  [1 total]

I.  The dodecahedron may have one black face (1).  [Running total - 2]

II.  The dodecahedron with two black faces might have them opposite one another (1), connected by an edge (1) or separated by one other face (1).  [Running total - 5]

**At this point I find it easier to describe everything in relationship to a side that is face down on the table. The faces adjacent to the side on the table form the first row, and the faces adjacent to the first row form the second row**

III.  The dodecahedron with three black faces might have:

  A.  All three touching each other:
    1.  One black face to the table - two adjacent to the first and touching each other (common vertex) (1)
    2.  One black face to the table - two adjacent to the first and not touching each other (1)

  B.  Two touching
    1.  One black face to the table - one adjacent - third opposite the table (1)
    2.  One white face to the table - two adjacent to the first and touching each other - third black face adjacent to the table but not adjacent to  the first two.  (1)

  C.  None touching each other.
     1. One black face to the table.  The second and third go into the second row but not adjacent to each other.  (1) [running total - 10]

IV.  The dodecahedron with four black faces might have:

  A.  All four touching each other
    1.  One black face to the table - three adjacent to the first and next to each other in the first row of faces touching the table.  (1)
    2.  One black face to the table - three adjacent to the first - all in the row touching the table two next to each other on one side one on the opposite side.  (1)
    3.  One white face to the table - four adjacent to the table (1)
    4.  One white face to the table - three adjacent to the table and connected in a row.  Fourth in the next row but adjacent to the black pentagon on the left.  (1)
    5.  One white face to the table - three adjacent to the table and connected in a row.  Fourth in the next row but adjacent to the black pentagon on the right.  (1) (This is different from the previous due to orientation)

  B.  Three touching each other
    1.  One black face to the table - two adjacent to the table and next to each other
       a.  Fourth black face opposite the table (1)
    2.  One white face to the table - three adjacent to the table and next to each other
       a.  Fourth black face opposite the table (1)
       b.  Fourth black face opposite center of the first three black faces (1).

  C.  Two touching each other
    1.  One black face to the table - one adjacent to the table    These two have six faces that are adjacent to one or both, leaving four available faces to place the remaining two:
       a.  Two black faces opposite the first two (1)
       b.  Two black faces filling the available positions not opposite the first two (these will not touch each other) (1)
       c.  The third black face opposite the table, the fourth not opposite the second (nor adjacent to the first two) (1)  [Running total 21]

Edited on March 4, 2006, 7:59 pm
  Posted by Leming on 2006-03-04 19:47:13

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