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 5×45 \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.)

Polyomino Tiling

           
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