We are given a flag with 7 regions. We want to color the flag such that no two touching regions have the same color. We want to find the minimum number of colors needed.

1. Problem Description

We are given a flag with 7 regions. We want to color the flag such that no two touching regions have the same color. We want to find the minimum number of colors needed.

2. Solution Steps

Let's analyze the flag and determine which regions are touching each other. We can label the regions from top to bottom and left to right:

1. Top left region

2. Top right region

3. Middle left region

4. Middle right region

5. Bottom left region

6. Bottom middle region

7. Bottom right region

The touching regions are:

- 1 touches 3

- 2 touches 4

- 3 touches 1, 5, 6

- 4 touches 2, 6, 7

- 5 touches 3, 6

- 6 touches 3, 4, 5, 7

- 7 touches 4, 6

We can try to color the flag with 2 colors. However, region 6 touches 4 other regions (3, 4, 5, 7). If these 4 regions were colored with the same color, then region 6 would require a different color, and vice versa. So we can investigate how many regions can be colored with one color.

Consider coloring region 6 with color A. Then regions 3, 4, 5 and 7 must be colored with a different color, say color B.

Region 1 touches region 3, so if region 3 has color B, region 1 must have color A.

Region 2 touches region 4, so if region 4 has color B, region 2 must have color A.

Therefore, we would need 2 colors.

However, notice region 6 is adjacent to the regions 3, 4, 5, and

7. Consider the situation where regions 3, 4, 5 and 7 are all different colors. This is not possible.

Suppose regions 3, 4, 5, and 7 can be colored with only two colors, which are A and B. Say 3, 5 have color A, and 4, 7 have color B. Then 1 must have color B (since 3 has color A), and 2 must have color A (since 4 has color B). Then 6 must be a third color, C. Thus, at least 3 colors are needed.

Let's see if 3 colors are sufficient. Color the top left and right regions with color

1. Color the two middle regions with color

2. Color the three bottom regions with color 3, 1, and 3 respectively, in a left-to-right fashion. Then we would need 3 colors.

Since at least 3 colors are needed and 3 colors is a viable solution, the minimum number of colors needed is

3.

3. Final Answer

3