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14 Sep 2006, 03:22
Kudos
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The above drawing is of a wooden paperweight that will be manufactured so that each of the 9 edges is 5 cm long. Originally, each side was to be painted either red, blue or yellow. However, the pallette has been expanded to include green. This expansion means that the number of distinguishable paperweights that can be made has increased by approximately r%, where r=
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14 Sep 2006, 10:34
Kudos
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My shot at this (definately took me more than 2 mins)...
We have 2 triangles and 3 squares.
{2T, 3S} = 5 faces.
Case 1: 3 Colors
Triangles:
Both faces the same color= 3
Both faces different color = 3x2/2 = 3
Total number of distinguishable colors = 6
Squares:
Each face different color = 1 ways
Two faces the same color = 6 ways
Three faces the same color = 3
Total number of ways = 10
Total distinguishable colorings = 10x6 = 60
Case 2: 4 colors
Triangles:
Both faces the same color = 4
Each face a different color = 4x3/2 = 6
Total = 10 ways
Squares:
Each face a different color = 4x3x2/3x2= 4
All faces the same color = 4
Two faces the same color = 6 ways
Total = 4+4+6 = 14
Total distinguishable colorings = 10 x 14 = 140
%tage increase = 140-60/60 = 80/60 = 4/3 = 133%
Therefore r = 133.
We have 2 triangles and 3 squares.
{2T, 3S} = 5 faces.
Case 1: 3 Colors
Triangles:
Both faces the same color= 3
Both faces different color = 3x2/2 = 3
Total number of distinguishable colors = 6
Squares:
Each face different color = 1 ways
Two faces the same color = 6 ways
Three faces the same color = 3
Total number of ways = 10
Total distinguishable colorings = 10x6 = 60
Case 2: 4 colors
Triangles:
Both faces the same color = 4
Each face a different color = 4x3/2 = 6
Total = 10 ways
Squares:
Each face a different color = 4x3x2/3x2= 4
All faces the same color = 4
Two faces the same color = 6 ways
Total = 4+4+6 = 14
Total distinguishable colorings = 10 x 14 = 140
%tage increase = 140-60/60 = 80/60 = 4/3 = 133%
Therefore r = 133.
