Question

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 $S?$

$72$

$76$

$80$

$84$

$88$

Solution:

The three equal sides of such a triangle must all have the same length. Checking the possible squared lengths in the $3 \times 3 \times 3$ grid, only three families of side occur.

Face diagonals of a unit cube (length $\sqrt2$ ): each of the $8$ unit cubes contributes $8$ triangles, one at each corner, for $8 \cdot 8 = 64.$

Face diagonals of the $2 \times 2 \times 2$ cube (length $2\sqrt2$ ): the three faces meeting at a vertex form one triangle, giving $8$ triangles.

Edge-midpoint segments (length $\sqrt6,$ joining midpoints of two edges): each of the $12$ edge midpoints is a vertex of two such triangles, for $\dfrac{12 \cdot 2}{3} = 8.$

The total is $64 + 8 + 8 = 80.$

Thus, the correct answer is C.

2005 AMC 12A Problem 25

Solution:

Problem 25 in Other Years