Geometry
# Isometries

There is a new company, Tile Carpets, that sells carpets as individual tiles that can be laid down side by side to form a single solid plane. These pieces of carpet are sold in sets, where every piece in the set is the same shape and size.

In order to form a single carpet, customers arrange the carpet tiles on the floor with no gaps in between them. This is possible for some shapes, but not for others. Which of the following shapes is impossible to lay down with no gaps?

**A Somewhat Squashed Star:**

Bob draws the following lines; however he is not sure if he is completely correct. Please help him.

Which of these lines are **not** lines of symmetry for the given figure? Find the sum of all the incorrect lines.

- As an explicit example, if you think lines \(2\) and \(3\) are not lines of symmetry, then your answer is \( 2 + 3 = 5\).

A square dance has the following commands:

- "Rotate Right" means everyone moves one pace around the square, counterclockwise.
- "Rotate Left" means everyone moves one pace around the square, clockwise.
- "Diagonal Swap" means the person in the lower-left position and the person in the upper-right position switch places, while the other two people stay still.

The Butterfly Basket is a dance that consists of many moves that individual do dancing in place, plus **21 repetitions** of the sequence:

**{Diagonal Swap -- Rotate Left -- Rotate Left -- Diagonal Swap -- Rotate Right -- }**

If Amy starts the dance in the upper left corner of her square, where does she end up at the end of the Butterfly Basket?

In the above image, how many lines of symmetry does the snowflake have?

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