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Tiling Rectangles with Polyominoes

by Henri Picciotto, The Urban School of San Francisco

Some polyominoes can be used to tile rectangles. For example, here is the smallest rectangle that can be tiled by the bent tromino:

two by three
  1. For each of the (non-rectangular) tetrominoes, i.e. the l, the n, and the t, can it be used to tile a rectangle? If yes, what is the smallest possible such rectangle? Show each tiling on graph paper.
  2. The L, P and Y pentominoes can each tile a rectangle. What is the smallest possible such rectangle? Show each tiling on graph paper.
  3. Tile a 3 by 5 rectangle with:
    1. U and X
    2. V and Z
  4. Tile a 4 by 5 rectangle with:
    1. T and Y
    2. U and N
    3. V and F
    4. V and N
  5. Tile a 5 by 5 square with:
    1. X and Y
    2. Y and Z
    3. Y and F
    4. L and X
  6. Tile a 3 by 10 rectangle with:
    1. U and Y
    2. U and F
  7. Find the smallest rectangle that can be tiled with
    1. Y and N.
    2. T and N.
    3. T and W.

This figure shows a rectangle tiled with a single F pentomino, and many P's:

F and P's
  1. What is the smallest rectangle you can tile with a single one of each of the pentominoes, and as many P's as you want?