31. Cube. Holding a model of a cube in your hand so that it can rotate around its longest axis (i.e., around the line connecting opposite vertices), you can wind black yarn around it without any gaps. The yarn will shade only half of the cube (why?). The same can be done with another axis; there are four in total, and each time we use a different color of yarn (black, red, blue, and
yellow). The entire model will be covered with different colors, and from their mixing, mixed colors will emerge (the model of the cube is white and we do not consider this color). How many color shades will there be on the cube and which ones?
31. The yarn wound around a cube rotating about one of its axes (Fig. 41) will remain only on those edges that do not have common points with the axis of rotation; the yarn will cover half of each face of the cube, i.e., half of the cube's surface.

Now, let's rotate the cube sequentially around each of the four axes, each time winding yarn of a different color.

Thus, when rotating around the axis $A C^{\prime}$, we will use yarn of color $a$ (black); when rotating around the axis $D B^{\prime}$, we will use yarn of color $b$ (red); when rotating around the axis $B D^{\prime}$, we will use yarn of color $c$ (yellow); and finally, when rotating around the axis $C A^{\prime}$, we will use yarn of color $d$ (blue).

The cube will be colored as shown in Fig. 42, where each face is divided into four triangles; the letters placed in the corresponding triangles denote the colors of the yarn covering that triangle.

It is easy to notice that:

$1^{\circ}$ six shades will appear on the surface of the cube, i.e., as many as there are combinations of four elements taken two at a time, namely: $a b, a c, a d, b c, b d$, and $c d$;

$2^{\circ}$ each face will have four different shades;

$3^{\circ}$ the surface of the cube will be covered by two layers of yarn;

$4^{\circ}$ opposite faces of the cube will be colored in the same shades, arranged in reverse cyclic order.

*) See, for example, I. M. Yaglom, Geometric Transformations II, Moscow, Gostekhizdat, 1956, §§ 1 and 4 of Chapter II; H. S. M. Coxeter, Introduction to Geometry, Chapter 6; G. Rademacher, O. Toeplitz, Numbers and Figures, Topic 18; D. Hilbert, S. Cohn-Vossen, Anschauliche Geometrie, Moscow, Gostekhizdat, 1951, § 36.

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-083.jpg?height=685&width=597&top_left_y=571&top_left_x=364)

Fig. 41.