7. What is the smallest (same) number of pencils that need to be placed in each of 6 boxes so that in any 4 boxes there are pencils of any of 26 predefined colors (there are enough pencils available)?
Answer: 13.

Solution. On the one hand, pencils of each color should appear in at least three out of six boxes, since if pencils of a certain color are in no more than two boxes, then in the remaining boxes, which are at least four, there are no pencils of this color. Therefore, the total number of pencils should be no less than $3 \cdot 26$.

On the other hand, if we take 3 pencils of each of the 26 colors and arrange them so that no pencils of the same color are in the same box, then a pencil of any color will be found in any four boxes (since there are only two other boxes). The required arrangement of pencils can be achieved, for example, by dividing all the pencils into 3 groups of 26 pencils of all colors and distributing each group evenly (13 pencils) into two boxes.