2005 AMC 12A Problems/Problem 25

Problem

Let be the set of all points with coordinates , where , , and are each chosen from the set . How many equilateral triangles all have their vertices in ?

Solution

Solution 1 (non-rigorous)

For this solution, we will just find as many solutions as possible, until it becomes intuitive that there are no more triangles left.

Take an unit cube. We try to make three of its vertices form an equilateral triangle. This we find is possible by taking any vertex, and connecting the three adjacent vertices into a triangle. This triangle will have a side length of ; a quick further examination of this cube will show us that this is the only possible side length. Each of these triangles is determined by one vertex of the cube, so in one cube we have 8 equilateral triangles. We have 8 unit cubes, and then the entire cube, giving us 9 cubes and equilateral triangles.

It may be tempting to connect the centers of the faces and to call that a cube, but a quick look at this tells us that that figure is actually an octahedron.

Now, we look for any additional equilateral triangles. Since the space diagonals of the cube cannot make an equilateral triangle, we will assume symmetry in the cube. A bit more searching shows us that connecting the midpoints of three non-adjacent, non-parallel edges gives us more equilateral triangles.

unitsize(0.3inch);
import three; 
currentprojection=perspective(5,4,3); pointpen = black; pathpen=black; pen l = linewidth(0.5); pen d = linewidth(0.8);
int i,j,k;
for(i=0;i<=2;++i)
 for(j=0;j<=2;++j)
  for(k=0;k<=2;++k)
   dot((i,j,k));
/* draw in cube edges */
D((0,0,0)--(0,0,2)--(0,2,2)--(0,2,0)--(0,0,0)--(0,2,0)--(2,2,0)--(2,0,0)--(0,0,0)--(2,0,0)--(2,0,2)--(0,0,2)--(0,0,0),l);
D((2,2,2)--(0,2,2)--(0,2,0)--(2,2,0)--(2,2,2)--(2,2,0)--(2,0,0)--(2,0,2)--(2,2,2)--(2,0,2)--(0,0,2)--(0,2,2)--(2,2,2),l); 
/* draw in equilateral triangle */
D((1,0,0)--(2,2,1)--(0,1,2)--(1,0,0),d);
/* darken points: D((1,0,0)^^(2,2,1)^^(0,1,2),d); */
 (Error making remote request. Unknown error_msg)

Notice that picking these three edges leaves two vertices alone, and that picking any two opposite vertices determine two equilateral triangles. Hence there are additional equilateral triangles, for a total of .

Solution 2 (rigorous)

The three dimensional distance formula shows that the lengths of the equilateral triangle must be , which yields the possible edge lengths of

Some casework shows that are the only lengths that work, from which we can use the same counting argument as above.

See Math Jam solution.

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