How many distinct triangles can be constructed by connecting three different vertices of a cube?

Source: mathcontest.olemiss.edu 11/1/2010

**Solution
**

**STEP 1** Try something simpler first.

Consider a square in a plane (it doesn’t look square because we look at it from space). Connect vertex to vertex and we have constructed distinct triangles and . Similarly, connect vertex to vertex to form more distinct triangles and . Thus, with vertices we can construct distinct triangles. We need vertices to make a triangle. So how many ways can we choose different vertices out of vertices? The answer is .

**STEP 2** Now try a square and one point in space above the square.

By looking carefully at the above figure, we can see that there are a total of distinct triangles:

Alternatively, we can think of choosing different vertices out of vertices to build our distinct triangles, .

**STEP 3** One square and two points and in space above the square.

It is getting much harder to count the triangles visually, so we are going to use a different method to identify them. We know that there are triangles. Let’s write down the vertices in alphabetical order and without looking at the figure pick out different vertices at a time as follows.

List the vertices in alphabetical order

pick out different vertices at a time

Now, look back at the above figure and verify to your satisfaction that indeed there are exactly distinct triangles.

**STEP 4** Now consider a cube.

How many distinct triangles can we construct by joining three different vertices?

From the patterns we have seen before, we know that there are distinct triangles. We can verify this fact by looking at the figure of the cube and list out the triangles as follows.

Front face ABCD | ABC | ABD | ACD | BCD |

Rear face EFGH | EFG | EFH | EGH | FGH |

Top face BCFG | BCF | BCG | BFG | CFG |

Bottom face ADEH | ADE | ADH | AEH | DEH |

Left face CDEF | CDE | CDF | CEF | DEF |

Right face ABGH | ABG | ABH | AGH | BGH |

Front face ABCD and vertex E | ABE | ACE | BCE | BDE |

Front face ABCD and vertex F | ABF | ACF | ADF | BDF |

Front face ABCD and vertex G | ACG | ADG | BDG | CDG |

Front face ABCD and vertex H | ACH | BCH | BDH | CDH |

Rear face EFGH and vertex A | AEF | AEG | AFG | AFH |

Rear face EFGH and vertex B | BEF | BEG | BEH | BFH |

Rear face EFGH andvertex C | CEG | CEH | CFH | CGH |

Rear face EFGH and vertex D | DEG | DFG | DFH | DGH |

Be mindful that there are duplicate triangles that were deleted from the above list. For example, in the case of front face and vertex , there are duplicate triangles out of triangles. They are listed underlined as follows:

Also, there is no need to consider the left and right faces and their opposite vertices, because the triangles will be duplicated.

Thus, we can construct 56 distinct triangles by joining three different vertices of a cube.