### Beautiful Geometry

These puzzles all involve polyominoes, shapes constructed by attaching two or more congruent squares side-by-side.

The shape above is a pentomino because it uses 5 squares, but any number of squares is possible. In the puzzles ahead, you'll fit them together into shapes and patterns like the tesselation below!

# Polyomino Tiling

Using only copies of the polyomino on the left (rotations allowed), is it possible to fill the shape on the right without overlapping or gaps?

# Polyomino Tiling

The type of puzzle you just did is called a tiling. You can assume a tiling will always cover an entire shape and never have any gaps or overlaps.

If one of the squares marked with a letter is removed, the shape on the right can be tiled by the polyomino on the left. Which square should be removed?

# Polyomino Tiling

For two of the three tetrominoes on the left, it's possible to use 4 copies of that tetromino (with rotation allowed) to tile a 4 by 4 square.

One of the tetrominoes will not be able to tile the square. Which one?

# Polyomino Tiling

If I have the 5 colored shapes shown that I can rotate, and I use each shape once, is it possible to place them so they fit perfectly in a $5 \times 4$ rectangle?

(The checkerboard pattern is a hint!)

# Polyomino Tiling

The three pentominoes on top can be used to tile one or both of the larger shapes. Which one(s)? (Pieces can be rotated or reflected; all three pentominoes must be used on a given tiling.)

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