How many different colored tetrahedra are there in which each face is colored ei

Question

How many different colored tetrahedra are there in which each face is colored either or white or blue? How many different colored cubes are there in which each face is colored either red or white or blue? How many different pinwheels with 6 identically shaped pins are there, if each pin can be colored one of 4 different colors? How many different pinwheels with 8 identically shaped pins are there, if each pin can be colored one of 4 different colors? Instead of pinwheels, consider bracelets with 8 identical size spherical beads, each of one of 4 different colors? How many different bracelets are there? [Now reflectional symmetries must be considered, in addition to rotational symmetries.]

Explanation / Answer

For colouring the faces a regular tetrahedron with n colours

Then number of tetrahedron=(11n2+n4)/12

=(11*9+81)/12=180/12=15

We know that tetrahedron has four faces out of which 3 faces are coloured

We can rotate tetrahedron by 12 ways in three direction 0,180,120

In 120 we can rotate either way about an axis passing through a vertex and the center of the opposite face, 8 of those; or 180o if rotation is 120 about an axis through a vertex v: for the coloring to be invariant about an axis with n colours there will be n2 different invariants

So total you there will 11n2 different tetrahedrons for 120 and 0 rotation

Now consider case 180 rotation you have n4 different patterns

For n faces colored we have(11n2+n4)/12 different tetrahedrons

Related Questions

Navigate

• Browse (All)
• Browse H
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.