Rules and solutions
Colorikub
The puzzle, also called The Six-Colour Cubes, was created by Percy Alexander MacMahon, a renowned English mathematician in combinatorics, and made its first appearance in 1921 in his work New Mathematical Pastimes .
1. The cubes
Here are 30 small cubes. Each cube has a unique combination of 6 colors. This means that no two cubes are alike.
Definitions
Trick
Unless you have an exceptional memory, you will often be called upon to recheck the cubes placed, so it is best to leave 2 to 3 cm between the cubes before final assembly.
2. The anticube
The simplest of all cube problems. Choose a cube C at random and find its mirror image. Each cube in the game has its anticube.
Once this cube is found and, whatever the face in contact C¹ or C² the anticube is present.
3. The 6 x 1 rectangle
The conditions of this game are met if:
4. The 3 x 3 x 3 cube
Choose any cube as a model. Reproduce its color combination by forming a 3 x 3 x 3 cube. However, in the internal structure, none of the 6 faces of a cube should be in contact with a face of the same color.
5. The pyramid
The reproduction of this famous monument obeys the same anti-domino rules as the 3 x 3 x 3 cube except that the color of the support face is random. The game can lend itself to elegant solutions by taking care of the checkerboard of the first course. The base of a cube of a higher course must be different from the 2, 3, or 4 colors that it is called upon to cover.
The base course can be a reserve to cope with unsuitable cubes when progressing to the upper courses.
6. The Colorful Tower
Experience gained from previous games allows this more demanding game to be handled better. The goal is to build a tower with a 2 x 2 base using all the cubes.
All internal faces in contact are of the same color. On the other hand, on the sides of the Tower we should not see two faces of the same color in contact.
This scheme would be ideal, but in practice we cannot avoid, at least in one case, the pairing of two shades. This problem, if it is invisible on the outside, necessarily acts on internal faces. That said, it is not proven that the case is insoluble.
7. The 2 x 2 x 2 cube
Choose a cube at random and reproduce its color combination by forming a 2 x 2 x 2 cube. At first glance, it is difficult to see the complexity of this exercise, but it lies in this diabolical constraint: All the internal faces must be in contact with faces of the same color.
8. The rectangle 4 x 2 x 2
Subject to the same constraints as the previous amusement, this parallelepiped is its mirror.
3. solutions
The 2 x 2 x 2 and 4 x 2 x 2
In the following example, we must start by finding the cube identical to the model below. We see that the opposite faces are blue-black, yellow-green and red-orange.
From the remaining 29 cubes, let's eliminate all cubes with a pair of the same kind. We thus select 16 cubes. Some models may only have 15, so we must change the model.
Let's take the model between our fingers. Aiming at point A on the blue face, we select two cubes whose visible faces are red/blue/yellow. We do the same with point B , (red/black/green). Then we turn the model over, aiming at points C and D on the orange face, also taking two cubes from each.
