Can someone help me with this question i've spent hours on it!!:(

6. Eight square tiles are laid out on a table so that they make a solid pattern. Each tile must touch at least one other tile along an entire edge. The squares all have sides of length 1.

Here is an example: The figures have a perimeter of 14 and 16 respectively.

(a) What are the possible perimeters of the figures that can be formed?

(b) Which figure has the smallest perimeter?

(c ) Which figure has the largest perimeter?

Sure! Let's break down this question step by step to find the answers.

(a) To find the possible perimeters of the figures that can be formed using the eight square tiles, we need to consider the different ways the tiles can be arranged.

One way to approach this is by starting with the smallest possible perimeter and progressively adding tiles to see how the perimeter changes. Let's first consider the scenario where all eight tiles are arranged in a straight line:

_ _ _ _ _ _ _ _

In this case, we have eight tiles lined up consecutively. Each tile touches two other tiles, except for the two end tiles which touch only one. So the perimeter in this scenario is 8 + 1 + 1 = 10.

Next, let's consider the scenario where the tiles are arranged in a square:

_ _ _ _
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| |
_ _ _ _

In this case, each tile touches four other tiles along its four edges. So the perimeter in this scenario is 4 + 4 + 4 + 4 = 16.

Now, let's explore other possible arrangements of the eight tiles. Some may have more complex patterns, but as long as each tile is touching at least one other tile along an entire edge, they are valid.

By continuing this process, you can try arranging the tiles in different ways and calculate the corresponding perimeters. Keep in mind that there may be multiple valid arrangements and resulting perimeters.

(b) To determine which figure has the smallest perimeter, you will need to calculate the perimeters for all the possible figures you found in part (a) above and compare them. The figure with the smallest perimeter is the one with the lowest calculated value.

(c) Similarly, to identify which figure has the largest perimeter, you will need to calculate the perimeters for all the possible figures found in part (a) and compare them. The figure with the largest perimeter is the one with the highest calculated value.

As you go through the process, you may find it helpful to draw diagrams or use a grid paper to visualize the different tile arrangements. This will make it easier to calculate the perimeters accurately.

I hope this explanation helps you solve the problem! Let me know if you have any further questions.