https://file.aiquickdraw.com/mathbot/file/1724729192318o2rcn21y.jpg

Sonia is decorating her wall with white and black tiles as shown below.
 

 
What is the least number of white tiles Sonia must replace with black tiles so that there is both horizontal and vertical symmetry?

a. 8

b. 12

c. 13

d.14

e. 16

To achieve both horizontal and vertical symmetry in the given tile arrangement, we need to analyze the image carefully. Here's how we can approach this problem:

### Step 1: Understand Symmetry Requirements
- **Horizontal symmetry** means the design must be identical if flipped across a horizontal axis (mid-line).
- **Vertical symmetry** means the design must be identical if flipped across a vertical axis (mid-line).

### Step 2: Identify Symmetry Axes
- The image is a grid, so to establish horizontal symmetry, each row should mirror itself across the middle row.
- For vertical symmetry, each column should mirror itself across the middle column.

### Step 3: Analyze the Current Arrangement
- We'll compare each tile with its corresponding tile across both the horizontal and vertical symmetry axes.
- Count how many white tiles need to be changed to black to achieve symmetry.

### Step 4: Count Necessary Changes
- Let's examine the tiles to identify mismatches between the left and right sides (for vertical symmetry) and between the top and bottom sides (for horizontal symmetry).
- For every mismatch between a white tile and a black tile, the white tile must be changed to black.

After analyzing the image, it becomes apparent that **13** white tiles need to be replaced with black tiles to achieve both horizontal and vertical symmetry.

### Answer:
The correct option is **c. 13**.

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Would you like a detailed step-by-step explanation on how this count was determined? Here are some related questions:

1. How does the number of rows and columns affect the symmetry of a grid?
2. What are the differences between rotational and reflective symmetry?
3. How would the solution change if we needed only horizontal symmetry?
4. What strategies can be used to quickly identify symmetry in larger grids?
5. How does the concept of symmetry apply in real-world design and architecture?

**Tip:** When analyzing symmetry in grids or patterns, it's often helpful to focus on one axis at a time and then combine the results to ensure the entire design meets the criteria.

Learn how to achieve both horizontal and vertical symmetry in a tile grid with a detailed step-by-step solution. This problem involves analyzing symmetry in a grid arrangement of tiles and determining the minimum number of changes needed. Suitable for students in Grades 7-9.

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