Question 1

First, we find the total number of white squares and the size of each region.

Step 1 — Calculate Region Size

The grid has 5 rows and 5 columns. So, the total number of squares in the grid is $5 \times 5$.

$\text{Total squares} = 5 \times 5$

$= \mathbf{25}$

One square in the center is green. So, we subtract the green square from the total squares to find the white squares.

$\text{White squares} = 25 - 1$

$= \mathbf{24}$

We need to divide these 24 white squares into 6 congruent regions. Congruent means they must have the exact same shape and size. Let us find the number of squares in each region.

$\text{Squares per region} = \frac{\text{Total white squares}}{\text{Number of regions}}$

$= \frac{24}{6}$

$\boxed{\mathbf{4 \text{ squares}}}$

Each of the 6 regions must be made of 4 white squares and must have the same shape.

Step 2 — Draw the Regions

We need to find a shape made of 4 squares that can tile the remaining 24 white squares into 6 identical pieces. Let us try an L-shaped region. This shape is a $2 \times 2$ square with one corner square removed. There are two orientations for this L-shape, and we can rotate them.

Let's label the rows from 1 to 5 (top to bottom) and columns from 1 to 5 (left to right). The green square is at (3,3).

Consider the following L-shaped region: Region 1: (1,1), (1,2), (2,1), (2,2) - This is a 2x2 square. This will not work as it leaves a central area that cannot be divided into 2 congruent L-shapes.

Let's try a different L-shape. Consider the shape formed by squares (1,1), (2,1), (2,2), (3,2). This is an L-tetromino. Let's mark this as Region 1.

X . . . . X X . . . . X G . . . . . . . . . . . .

Now, let's find 5 more regions that are congruent to this one and cover all remaining white squares without overlapping. We can use rotation and reflection to find the other regions.

Let's define the 6 regions:

1. Region 1 (Top-Left): (1,1), (2,1), (2,2), (3,2)
2. Region 2 (Top-Right): (1,5), (2,5), (2,4), (3,4)
3. Region 3 (Bottom-Left): (5,1), (4,1), (4,2), (3,2) - Wait, (3,2) is used by Region 1. This means this specific L-shape does not work with simple rotation/reflection around the center.

Let's try another L-shape. Consider the shape formed by squares (1,1), (1,2), (2,2), (2,3). This is an L-tetromino. Let's mark this as Region 1.

X X . . . . X X . . . . G . . . . . . . . . . . .

Let's rotate this shape by 90 degrees clockwise around the center of the grid (which is the center of the green square (3,3)). Region 2: (1,5), (2,5), (2,4), (3,4) Region 3: (5,5), (5,4), (4,4), (4,3) Region 4: (5,1), (4,1), (4,2), (3,2)

These 4 regions are congruent and use $4 \times 4 = 16$ white squares. Let's see the remaining white squares: (1,3), (1,4) (2,1) (3,1), (3,5) (4,5) (5,2), (5,3)

This remaining shape is not easily divisible into two congruent 4-square regions. So this L-shape also does not work.

The correct shape is an L-tetromino that looks like a $2 \times 2$ square with one corner removed. Let's try to place them carefully.

Consider the following L-shaped region: Region 1: (1,1), (1,2), (2,1), (3,1) X X . . . X . . . . X . G . . . . . . . . . . . . This is an L-tetromino. Let's call it L1.

Now, let's place other regions. Region 2: (1,4), (1,5), (2,5), (3,5) . . . X X . . . . X . . G . X . . . . . . . . . . This is L1 rotated.

Region 3: (5,1), (5,2), (4,1), (3,1) - This overlaps with Region 1 at (3,1). This won't work.

The key is to use a specific L-tetromino shape that allows for tiling. Let's try the shape: X X X . X . This is not a standard L-tetromino. It is a 1x3 strip with an extra square attached to the middle. This is a T-tetromino.

Let's try the L-tetromino shape: X X X X This is a T-tetromino.

Let's try the L-tetromino shape: X X . X . X This is an L-tetromino. Let's try to place it. Region 1: (1,1), (1,2), (2,2), (3,2) X X . . . . X . . . . X G . . . . . . . . . . . . Let's rotate this by 90 degrees clockwise around the center (3,3). Region 2: (1,5), (2,5), (2,4), (3,4) Region 3: (5,5), (5,4), (4,4), (3,4) - This overlaps with Region 2 at (3,4). This won't work.

Let's try a different approach. We have 24 white squares. Each region has 4 squares. Let's consider the central 3x3 block. It has 8 white squares: (2,2), (2,3), (2,4), (3,2), (3,4), (4,2), (4,3), (4,4). These 8 squares can form two congruent regions of 4 squares. Region A: (2,2), (2,3), (3,2), (4,2) Region B: (2,4), (3,4), (4,4), (4,3) These two regions are congruent L-tetrominoes.

. . . . . . A A B . . A G B . . A B B . . . . . . (A and B are parts of the two central regions)

Now we have $24 - 8 = 16$ white squares remaining. We need to form $6 - 2 = 4$ more congruent regions of 4 squares each. The remaining 16 squares are the outer frame. Can we form 4 congruent regions from the outer frame, which are also congruent to Region A and B?

The outer frame consists of: Row 1: (1,1), (1,2), (1,3), (1,4), (1,5) Row 2: (2,1), (2,5) Row 3: (3,1), (3,5) Row 4: (4,1), (4,5) Row 5: (5,1), (5,2), (5,3), (5,4), (5,5)

Let's try to draw the solution directly. The solution involves dividing the grid into 6 L-tetrominoes. Let's define the L-tetromino as a $2 \times 2$ square with one corner removed. Let's use the shape: XX X .

Region 1: (1,1), (1,2), (2,1), (3,1) 1 1 . . . 1 . . . . 1 . G . . . . . . . . . . . . Region 2: (1,4), (1,5), (2,5), (3,5) . . . 2 2 . . . . 2 . . G . 2 . . . . . . . . . . Region 3: (5,1), (5,2), (4,1), (3,1) - This overlaps with R1 at (3,1). This is not correct.

Let's try the L-tetromino shape: X X X X

This is a T-tetromino. Let's try to place it. Region 1: (1,3), (2,3), (2,2), (2,1) . . 1 . . 1 1 1 . . . . G . . . . . . . . . . . . Region 2: (3,5), (3,4), (2,4), (1,4) . . . 2 . . . . 2 2 . . G 2 . . . . . . . . . . . Region 3: (5,3), (4,3), (4,4), (4,5) . . . . . . . . . . . . G . . . . 3 3 3 . . 3 . . Region 4: (3,1), (3,2), (4,2), (5,2) . . . . . . . . . . 1 1 G . . . 4 . . . . 4 . . . These 4 regions are congruent and disjoint. They use 16 squares. The remaining 8 squares are: (1,1), (1,2), (1,5) (2,5) (4,1) (5,1), (5,4), (5,5)

This remaining shape is not symmetric and cannot be divided into two congruent regions of 4 squares each.

The solution is to use a specific L-tetromino. Let's try to draw the final solution. The regions are L-shaped, but they are not the standard