MathNet 2021 — Part 08 Question 81

There are $2021$ colors and $2021$ chips of each color. Arrange the $2021^2$ chips in a row. A chip $F$ is called “bad” if, on each side of $F$, there is an odd number of chips among the $2020\times2021$ chips whose color differs from that of $F$.

Part (a). Determine the smallest possible number of bad chips.

Part (b). If every chip is required to have at least one adjacent chip of the same color, determine the smallest possible number of bad chips.