3. We have 6 different colors and a large number of cubes. We paint each cube with all 6 colors, with each face painted one color. What is the maximum number of cubes we can paint so that each of them is painted differently? (If we can rotate one of the painted cubes so that the colors of the faces are arranged in the same way as on another cube, the cubes are considered to be painted the same.)
III/3. Denimo, da imamo na voljo rdečo, modro, zeleno, rumeno, oranžno in belo barvo. Kocke postavimo tako, da imajo spodnjo mejno ploskev pobarvano z belo barvo. Za zgornjo ploskev imamo na voljo katero koli izmed preostalih 5 barv.

Sedaj se vprašamo, koliko kock ima lahko zgornjo ploskev pobarvano z neko izbrano barvo (denimo oranžno). Te kocke postavimo tako, da imajo nepobarvano prednjo, zadnjo, levo in desno mejno ploskev. Z rumeno barvo pobarvamo zadnjo mejno ploskev posamezne kocke. Vsako izmed teh kock namreč lahko zavrtimo tako, da ima zadnjo ploskev rumeno, ne da bi pri tem spremenili barvo zgornje in spodnje ploskve. Preostale 3 mejne ploskve lahko pobarvamo na 6 načinov (npr. 3 barve imamo na voljo za desno mejno ploskev in 2 barvi za sprednjo mejno ploskev).

Vseh možnosti je torej $5 \cdot 6=30$. Pobarvamo lahko največ 30 kock, če želimo, da bo vsaka izmed njih drugačne barve.

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III/3. Suppose we have red, blue, green, yellow, orange, and white colors available. We place the cubes such that their bottom face is painted white. For the top face, we have any of the remaining 5 colors available.

Now we ask ourselves, how many cubes can have their top face painted with a certain color (for example, orange). We place these cubes such that their front, back, left, and right faces are unpainted. We paint the back face of each cube with yellow. Each of these cubes can be rotated so that the back face is yellow without changing the color of the top and bottom faces. The remaining 3 faces can be painted in 6 ways (for example, we have 3 colors available for the right face and 2 colors for the front face).

Therefore, the total number of possibilities is $5 \cdot 6=30$. We can paint a maximum of 30 cubes if we want each of them to have a different color.