Question 67
Lucio has a box with many 14-faced dice, all of the same shape and size but with different numberings. The shape of the dice is a truncated cube, that is, a cube whose vertices are cut off so as to form a polyhedron with 14 faces: 6 regular octagons and 8 equilateral triangles. Each die carries all the numbers from 1 to 14 on different faces, but in such a way that the sum of the numbers on opposite faces of the same die is always the same. However, for any two distinct dice there is no rotation taking the first one to the second. What is the largest possible number of dice in Lucio's box?