[CV117] Painting a Cube

117 - Painting a Cube

Painting a Cube is a puzzle in Professor Layton and the Curious Village.

Puzzle

In front of you sits a blank paper cube that you've decided to paint. You need to paint the cube so that all faces that touch are different colors.

Using three colors of paint, how many ways can you paint the cube so that it satisfies the above condition?

Each painting scheme should be a different pattern, not just the same pattern with the colors rearranged. Also, assume that you can't leave any sides of the square blank.

Hints

Click a Tab to reveal the Hint.

As you know, all cubes have six sides. Because of this structure, every face of the cube touches four others, meaning that only one of the five other faces doesn't touch any given face.

Taking Hint One a step further, in order to paint the cube three colors and have no two connecting faces be of the same color, you should use each color to paint opposing faces.

You need to paint two opposing faces of the cube each color. Count how many different ways there are of doing that and you've solved the puzzle.

Just remember, simply reconfiguring which colors go where doesn't count as an entirely new arrangement.

Solution

Incorrect

Nope!

All rotated and color-swapped versions of a particular painting scheme count as the same pattern. Don't count them as separate patterns when tallying up the total number of possible solutions.

Correct

If you have to paint the cube with three colors, then your only choice is to paint opposing sides of the cube the same color.

As seen in the diagram above, even if you were to change where you used each color, rotating the cube proves that you're really just reusing the same idea of painting opposing sides the same color.

There's only one unique way to color this cube using three paints.

v · d · ePuzzles in Professor Layton and the Curious Village

Normal	001 002 003 004 005 UK 006 007 008 009 010 UK JP 011 012 013 014 015 016 017 JP 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 JP 035 036 037 038 039 040 041 042 043 044 045 UK 046 UK 047 048 049 050 UK 051 052 053 054 055 056 057 058 059 060 061 062 UK 063 064 065 066 UK 067 UK 068 069 070 071 072 UK 073 074 075 076 UK 077 078 079 080 081 082 083 084 085 UK 086 087 088 UK 089 090 091 092 093 094 095 096 UK 097 098 099 100 101 102 103 104 105 106 107 108 109 110 111 UK 112 113 UK 114 115 116 117 118 119 120 121 122 123 124 UK 125 126 127 128 129 130 131 132 133 134 135

Weekly	See here for weekly puzzles

Puzzle Index: CV · DB · UF · LS · MM · AL · VS · LMJ