1. Two types of tiles were laid on the wall in a checkerboard pattern. Several tiles fell off the wall. The remaining tiles are shown in the picture. How many striped tiles fell off? Be sure to explain your answer.

![](https://cdn.mathpix.com/cropped/2024_05_06_27f0546c598e51c512d8g-1.jpg?height=414&width=782&top_left_y=730&top_left_x=640)
Answer: 15.

## Solution.

Method 1. Draw additional cells and count them.

![](https://cdn.mathpix.com/cropped/2024_05_06_27f0546c598e51c512d8g-1.jpg?height=411&width=780&top_left_y=1302&top_left_x=638)

Method 2. Look at the number of fallen cells by rows: in the second row from the top, 2 tiles fell out, of which 1 is striped, in the third row - 5 tiles, of which 2 are striped, in the fourth row - 7 tiles (3 striped) and so on. There are five rows in total, so the number of tiles that fell out is

$$
1+2+3+4+5=15 \text{ tiles. }
$$

Method 3. If the tiles had not fallen out, there would be 6 striped tiles in the odd rows and 7 striped tiles in the even rows on the original picture. Subtract the existing striped tiles from each row and add up the missing ones in each row.

$$
\begin{gathered}
(6-6)+(7-6)+(6-4)+(7-4)+(6-2)+(7-2)+(6-6)= \\
=0+1+2+3+4+5+0=15
\end{gathered}
$$

## Grading Criteria.

- Any correct solution - 7 points.
- A solution consisting of the line $1+2+3+4+5=15$ without explanations - 5 points.
- A correct approach in methods 2 or 3, but an incorrect answer due to an arithmetic error - 4 points.
- Only the correct answer (without the redrawn picture) - 2 points.