Question

Different shapes can be generated by sticking square tiles edge to edge. The figures below show examples of FOUR unique shapes generated by sticking FIVE square tiles in the given manner. Please note that shapes a, b, c and d of example 4 are NOT DIFFERENT from each other as any of them can be generated by simple operation(s) of rotation or flipping the others. 
What is the total number of unique shapes (including the four shown) that can be generated by sticking FIVE square tiles along their edges? 

Hint:

The number of polyominoes for a given number of squares is a standard combinatorial problem. It's useful to remember the first few counts: monomino (1), domino (1), trominoes (2), tetrominoes (5), and pentominoes (12).

Answer 1

Correct Answer: 12

Step 1: Understanding the Concept:
The problem asks for the total number of unique shapes that can be formed by joining five squares edge-to-edge. These shapes are known as pentominoes. The term "unique" here means that shapes that can be transformed into one another by rotation or reflection (flipping) are considered the same. These are technically called "free" pentominoes.
Step 2: Key Formula or Approach:
There is no specific formula to calculate this. The method is to systematically enumerate all possible arrangements of five squares. A good strategy is to start with a longer straight chain of squares and then try moving squares to different positions.
Step 3: Detailed Explanation:
Let's try to construct all possible pentominoes systematically.
1. 5 squares in a line: This forms the 'I' pentomino. (1 shape)
2. 4 squares in a line (as a base): Attach the fifth square to the side of this chain.
- Attaching to an end square forms the 'L' pentomino. (Flipping gives 'J', but they are the same free pentomino). (1 shape)
- Attaching to the second square from the end forms the 'N' pentomino. (Flipping gives 'S' or 'Z'). (1 shape)
- Attaching to one of the two middle squares forms the 'T' pentomino. (1 shape)
- Attaching to the other middle square also forms the 'T' pentomino.
We can also form the 'Y' pentomino from this base. (1 shape)
3. 3 squares in a line (as a base): Attach the remaining two squares.
- Attach both to one side of the middle square, one above and one below. This forms the 'X' pentomino. (1 shape)
- Attach a 2x1 block to the side of the middle square. This forms the 'P' pentomino. (1 shape)
- Attach a 2x1 block to the side of an end square. This forms the 'F' pentomino. (1 shape)
- Attach one square on top of an end square, and one on top of the other end square. This is not a valid shape as they are not edge-connected.
- Attach the two squares to form a corner on one end. This forms the 'U' pentomino. (1 shape)
4. Other configurations:
- A 'V' shape can be formed. (1 shape)
- A 'W' shape can be formed. (1 shape)
- A 'Z' shape (or 'S' if flipped) can be formed. (1 shape)
The 12 unique pentominoes are famously named after the letters of the alphabet they vaguely resemble: F, I, L, P, N, T, U, V, W, X, Y, Z.
The four shapes shown in the question are examples of these: 1 is 'I', 2 is 'X', 3 is 'L', and 4 shows four orientations of 'F'. The question asks for the total number of such unique shapes.
Step 4: Final Answer:
There are a total of 12 unique free pentominoes.