1. Each of the 4 colors has 6 pencils. Each of the six children received 4 pencils. For what smallest $k$ can we always choose $k$ children and decorate their pencils so that we have at least one in each color?
1. Let's denote colors by numbers from 1 to 4. If the distribution of pencils among children is

$$
(1,1,1,1), \quad(2,2,2,2), \quad(3,3,3,3), \quad(1,1,4,4), \quad(2,2,4,4), \quad(3,3,4,4)
$$

each child has only one of the colors $1,2,3$, so we need at least three children.

On the other hand, $k=3$ is sufficient: at least one child has pencils in two colors, and we will find the remaining two colors among the other two children.