Plan: We decide to find all the shapes that the pieces can be from a basis one.

Do: Because the hexagon is cut into 54 pieces so every piece we should found is made by 6 pieces.

We begin with a hexagon, made like this:

Then we pick out the triangle in the shadow and put it into area 1,2 or 3, we can get 3 shapes:

If we pick out two triangles and put them in different place we can get:

If we pick out three triangles and put them in different place we can get:

So we have 1(the basis one)+3+7+1=12 shapes.

Part 2: See: we find the most important information is: the pieces are used to cover the hexagon by eight different-shaped pieces with one piece being repeated to make nine pieces in all.

Plan: We decide to use all the shapes we found to do this problem.

Do: We try to find out all the answers and we have (the red two are repeated ones):

Check: We check our answers carefully we think we find all the answers.

When we aggregated all the solutions sent in, we found that we had received 26 different arrangements. Also,  some teams discovered that you don't actually need to repeat one of the pieces. As the Wolfram site explains, there are at least 15 different ways of tiling this T-hexagon with different hexiamonds. And there seem to be two ways of tiling it with just one hexiamond: