Exercise 12. In how many ways can the integers from 1 to 2021 be colored such that each integer is colored either blue, green, or red, and such that two consecutive integers are never the same color?
Solution to Exercise 12 There are three choices for the color of the number 1. Once the number 1 is colored, there are two choices for the color of the number 2, since the numbers 1 and 2 must be of different colors. The number 3 cannot have the same color as the number 2, so there are two possible choices for the color of the number 3.

More generally, after coloring the first $k$ numbers, when $k \geqslant 2$, there are two possible choices for the color of the number $k+1$, because the numbers $k+1$ and $k$ cannot be of the same color. In the end, there are therefore $3 \times 2^{2020}$ ways to color the integers.

Graders' Comments: This problem was generally well done. Be sure to multiply the possibilities for different independent portions rather than adding them. Many students tried to group the numbers. While this sometimes worked, it was not necessary. Other students made possibility trees for the first few integers. While this was not necessary, it was clearly a very good idea, as these reasonings almost always turned out to be excellent. We remind you that if you color a finite number of elements with a finite number of colors, you cannot end up with an infinite total number of possibilities. Some students tried to choose a particular color and distinguish the possible colorings according to the number of integers of that color. Unfortunately, this method was never conclusive.