2000 AMC 12 Problems/Problem 25

Problem

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)

Solution

We consider the dual of the octahedron, the cube; a cube can be inscribed in an octahedron with each of its vertices at a face of the octahedron. So the problem is equivalent to finding the number of ways to color the vertices of a cube.

Select any vertex and call it ; there are color choices for this vertex, but this vertex can be rotated to any of locations. After fixing , we pick another vertex adjacent to . There are seven color choices for , but there are only three locations to which can be rotated to (since there are three edges from ). The remaining six vertices can be colored in any way and their locations are now fixed. Thus the total number of ways is .

Though the cube may be easier to think about, the octahedron can be directly considered. Since the octahedron is indistinguishable by rotations, without loss of generality fix a face to be red.

There are ways to arrange the remaining seven colors, but there still are three possible rotations about the fixed face, so the answer is .

size(8cm);
defaultpen(0.5);
import three;
import math;
currentprojection=orthographic(2,0.2,1);
triple A=(0,0,1);
triple B=(sqrt(2)/2,sqrt(2)/2,0);
triple C=(sqrt(2)/2,-sqrt(2)/2,0);
triple D=(-sqrt(2)/2,-sqrt(2)/2,0);
triple E=(-sqrt(2)/2,sqrt(2)/2,0);
triple F=(0,0,-1);
picture p = new picture, r = new picture, s = new picture;
draw(p,A--B--E--cycle);
draw(p,A--C--D--cycle);
draw(p,F--C--B--cycle);
draw(p,F--D--E--cycle,dotted+linewidth(0.7));
fill(p,A--B--C--cycle,rgb(1,.6,.6));
fill(p,A--B--E--cycle,rgb(1,1,.6));
add(scale(2.2)*p);
draw(r,A--B--E--cycle);
draw(r,A--C--D--cycle);
draw(r,F--C--B--cycle);
draw(r,F--D--E--cycle,dotted+linewidth(0.7));
fill(r,A--B--C--cycle,rgb(1,.6,.6));
fill(r,A--C--D--cycle,rgb(1,1,.6));
add(scale(2.2)*shift(2*right)*r);
draw(s,A--B--E--cycle);
draw(s,A--C--D--cycle);
draw(s,F--C--B--cycle);
draw(s,F--D--E--cycle,dotted+linewidth(0.7));
fill(s,A--B--C--cycle,rgb(1,.6,.6));
fill(s,B--C--F--cycle,rgb(1,1,.6));
add(scale(2.2)*shift(4*right)*s);
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