2023 IOQM/Problem 7

Problem 7

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

Solution

We can fix 1 and 2 anywhere now we can permute the remaining faces like in a necklace (not a circle because anticlockwise and clockwise would be same), so the number of ways to permute them around the remaining 4 faces would be $\frac{3!}{2}$ and number of ways to colour them $2\cdot2\cdot2$ ways. So the answer would be $\boxed{\frac{3!}{2} \cdot8=\textbf{24}}$

~Lakshya Pamecha

Page

Toolbox

Search

2023 IOQM/Problem 7

Problem 7

Solution