## Task A-1.5.

In how many ways can we color the cells of a $2 \times 2016$ board in two colors so that there are no three cells of the same color that can be simultaneously covered by a tile of the shape shown in the figure? The tile can be rotated.

![](https://cdn.mathpix.com/cropped/2024_05_30_1a4376e76c5095e3db07g-06.jpg?height=149&width=140&top_left_y=1916&top_left_x=1552)
## Solution.

If both fields in the first column are white, then in the second column, no field can be white. Thus, both fields in the second column are black. Similarly, we conclude that both fields in the third column are white, and so on. This is one valid coloring.

Analogously, if both fields in the first column are black, we get another valid coloring.

![](https://cdn.mathpix.com/cropped/2024_05_30_1a4376e76c5095e3db07g-06.jpg?height=149&width=605&top_left_y=2421&top_left_x=654)

If the fields in the first column are of different colors, then in the second column, we cannot have two fields of the same color. Thus, the fields in the second column must also be of different colors, and both possibilities (the black field is in the first or second row) are allowed. The total number of colorings in this case is \(2 \cdot 2 \cdot \ldots \cdot 2 = 2^{2016}\).

## COUNTY COMPETITION IN MATHEMATICS

2nd grade - high school - A variant
March 3, 2016.

## IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE TASK, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH IN AN APPROPRIATE MANNER.