# 3. CONDITION

There is an unlimited number of chips in six colors. What is the smallest number of chips that need to be arranged in a row so that for any two different colors, there are two adjacent chips of these colors in the row?
Solution. From the condition, it follows that for each fixed color A, a chip of this color must be paired with a chip of each of the other 5 colors. In a row, a chip has no more than two neighbors, so a chip of color A must appear at least 3 times. Similarly for each other color. Thus, there should be no less than $3 \cdot 6=18$ chips. Here is one example of the desired arrangement of chips: 123456246325164135.

Answer. 18 chips.