Problem:
Let S be the set of all points with coordinates (x,y,z), where x,y, and z are each chosen from the set {0,1,2}. How many equilateral triangles have all their vertices in S?
Answer Choices:
A. 72
B. 76
C. 80
D. 84
E. 88
Solution:
Let A(x1​,y1​,z1​),B(x2​,y2​,z2​), and C(x3​,y3​,z3​) be the vertices of such a triangle. Let
(Δxk​,Δyk​,Δzk​)=(xk+1​−xk​,yk+1​−yk​,zk+1​−zk​), for 1≤k≤3
where (x4​,y4​,z4​)=(x1​,y1​,z1​). Then (∣Δxk​∣,∣Δyk​∣,∣Δzk​∣) is a permutation of one of the ordered triples (0,0,1),(0,0,2),(0,1,1),(0,1,2),(0,2,2),(1,1,1), (1,1,2),(1,2,2), or (2,2,2). Since △ABC is equilateral, AB,BC, and CA correspond to permutations of the same ordered triple (a,b,c). Because
k=1∑3​Δxk​=k=1∑3​Δyk​=k=1∑3​Δzk​=0
the sums
k=1∑3​∣Δxk​∣,k=1∑3​∣Δyk​∣, and k=1∑3​∣Δzk​∣
are all even. Therefore (∣Δxk​∣,∣Δyk​∣,∣Δzk​∣) is a permutation of one of the triples (0,0,2),(0,1,1),(0,2,2),(1,1,2), or (2,2,2).
If ( a,b,c)=(0,0,2), each side of â–³ABC is parallel to one of the coordinate axes, which is impossible.
If ( a,b,c)=(2,2,2), each side of △ABC is an interior diagonal of the 2×2×2 cube that contains S, which is also impossible.
If (a,b,c)=(0,2,2), each side of △ABC is a face diagonal of the 2×2×2 cube that contains S. The three faces that join at any vertex determine such a triangle, so the triple (0,2,2) produces a total of 8 triangles.
If (a,b,c)=(0,1,1), each side of â–³ABC is a face diagonal of a unit cube within the larger cube that contains S. There are 8 such unit cubes producing a total of 8â‹…8=64 triangles.
There are two types of line segments for which (a,b,c)=(1,1,2). One type joins the center of the face of the 2×2×2 cube to a vertex on the opposite face. The other type joins the midpoint of one edge of the cube to the midpoint of another edge. Only the second type of segment can be a side of △ABC. The midpoint of each of the 12 edges is a vertex of two suitable triangles, so there are 12⋅2/3=8 such triangles.
The total number of triangles is 8+64+8=80​.
The problems on this page are the property of the MAA's American Mathematics Competitions