10. 1. Distribute 40 pencils of four colors (10 pencils of each color) to 10 students, with each student getting 4 pencils. It is known that no matter how the pencils are distributed, there will always be $n$ students such that the $4 n$ pencils they have are of four colors. Find the minimum value of $n$.
10.1. The minimum value of $n$ is 3.

First, we prove: There are always 3 students who have pencils of 4 different colors.

Since there are 10 pencils of each color, and each student has 4 pencils, there must be a student who has at least two different colors of pencils.

Obviously, any other color of pencils must be owned by at least one student. Therefore, the conclusion holds.

Next, we provide an example to show that there exists a distribution method such that any two students have at most three colors of pencils.

Two students have 4 pencils of the second color, two students have 4 pencils of the third color, two students have 4 pencils of the fourth color, one student has 4 pencils of the first color, one student has 2 pencils of the first color and 2 pencils of the second color, one student has 2 pencils of the first color and 2 pencils of the third color, and one student has 2 pencils of the first color and 2 pencils of the fourth color.