This is an untested model in the same vein as my Base-32 Binary Poker Chips, which can count from 1 to 6 in octal with three-bit binary values on each face of the die. (The 000 and 111 values are unreachable: each face has at least one set bit and one unset bit.)

Features

The 1, 2, and 4 faces have a digit underneath demonstrating which set bit gives the value; the 3, 5, and 6 faces have lines underneath showing the two bits added together to reach the value.

Interestingly, due to the complementary nature of dice faces, all the bits on whatever face is rolled can be flipped by flipping the die itself.

Uses

This could be used for games that explain how numbers like chmod permission flags are set, in which the first, second, and third bits are each treated as a “flag” that gives the player permission to take some action.