It has long been known that only four pentominoes can tile rectangles:

For other rectangles that these pentominoes tile, see Mike Reid's Rectifiable Polyomino Page.

Rodoflo Kurchan's online magazine *Puzzle Fun*
studied the problem of tiling some rectangle with two different pentominoes,
in Issue
19, and revisited the problem
in Issue 21.
The August
2010 issue of
Erich Friedman's Math Magic broadened this problem
to use two polyominoes of any size, not necessarily the same.
My page Two-Pentomino Balanced Rectangles
shows rectangles tiled by two pentominoes in equal quantities.

Here I study the related problem of tiling some rectangle with three pentominoes, using the same number of copies of each.

For two pentominoes, see Two-Pentomino Balanced Rectangles.

F I L | 6 | F I N | 18 | F I P | 9 | F I T | 18 | F I U | 6 | F I V | 6 | F I W | 18 | F I X | 180 | F I Y | 12 | F I Z | 18 |

F L N | 6 | F L P | 6 | F L T | 6 | F L U | 6 | F L V | 6 | F L W | 6 | F L X | 30 | F L Y | 9 | F L Z | 12 | F N P | 12 |

F N T | 18 | F N U | 6 | F N V | 6 | F N W | × | F N X | × | F N Y | 12 | F N Z | × | F P T | 12 | F P U | 3 | F P V | 6 |

F P W | 12 | F P X | 42 | F P Y | 6 | F P Z | 18 | F T U | 18 | F T V | 24 | F T W | 24 | F T X | × | F T Y | 6 | F T Z | × |

F U V | 18 | F U W | 6 | F U X | 66 | F U Y | 12 | F U Z | 30 | F V W | 18 | F V X | 96 | F V Y | 12 | F V Z | 12 | F W X | × |

F W Y | 12 | F W Z | × | F X Y | 24 | F X Z | × | F Y Z | 12 | I L N | 6 | I L P | 6 | I L T | 12 | I L U | 6 | I L V | 6 |

I L W | 6 | I L X | ? | I L Y | 6 | I L Z | 12 | I N P | 6 | I N T | 6 | I N U | 6 | I N V | 6 | I N W | 18 | I N X | ? |

I N Y | 6 | I N Z | 12 | I P T | 6 | I P U | 6 | I P V | 6 | I P W | 9 | I P X | 24 | I P Y | 6 | I P Z | 6 | I T U | 21 |

I T V | 18 | I T W | 12 | I T X | 78 | I T Y | 6 | I T Z | 24 | I U V | 12 | I U W | 24 | I U X | 48 | I U Y | 6 | I U Z | 18 |

I V W | 12 | I V X | ? | I V Y | 6 | I V Z | 6 | I W X | 180 | I W Y | 6 | I W Z | 36 | I X Y | 18 | I X Z | ? | I Y Z | 12 |

L N P | 6 | L N T | 12 | L N U | 6 | L N V | 3 | L N W | 6 | L N X | 30 | L N Y | 6 | L N Z | 6 | L P T | 12 | L P U | 6 |

L P V | 3 | L P W | 6 | L P X | 24 | L P Y | 6 | L P Z | 6 | L T U | 18 | L T V | 6 | L T W | 18 | L T X | 6 | L T Y | 3 |

L T Z | 18 | L U V | 9 | L U W | 6 | L U X | 42 | L U Y | 6 | L U Z | 12 | L V W | 12 | L V X | ? | L V Y | 9 | L V Z | 6 |

L W X | 78 | L W Y | 6 | L W Z | 18 | L X Y | 18 | L X Z | ? | L Y Z | 6 | N P T | 6 | N P U | 3 | N P V | 6 | N P W | 12 |

N P X | 24 | N P Y | 6 | N P Z | 6 | N T U | 12 | N T V | 12 | N T W | 12 | N T X | 36 | N T Y | 6 | N T Z | 24 | N U V | 6 |

N U W | 24 | N U X | 18 | N U Y | 6 | N U Z | 6 | N V W | 6 | N V X | 78 | N V Y | 12 | N V Z | 6 | N W X | × | N W Y | 12 |

N W Z | × | N X Y | 15 | N X Z | × | N Y Z | 12 | P T U | 6 | P T V | 6 | P T W | 6 | P T X | 12 | P T Y | 6 | P T Z | 12 |

P U V | 3 | P U W | 12 | P U X | 6 | P U Y | 3 | P U Z | 6 | P V W | 6 | P V X | 30 | P V Y | 6 | P V Z | 6 | P W X | 36 |

P W Y | 6 | P W Z | 6 | P X Y | 12 | P X Z | 42 | P Y Z | 6 | T U V | 12 | T U W | 18 | T U X | 48 | T U Y | 6 | T U Z | 42 |

T V W | 30 | T V X | ? | T V Y | 18 | T V Z | 24 | T W X | 54 | T W Y | 6 | T W Z | 42 | T X Y | 24 | T X Z | × | T Y Z | 12 |

U V W | 60 | U V X | ? | U V Y | 12 | U V Z | 12 | U W X | 96 | U W Y | 6 | U W Z | × | U X Y | 6 | U X Z | 180 | U Y Z | 18 |

V W X | ? | V W Y | 12 | V W Z | 24 | V X Y | 42 | V X Z | ? | V Y Z | 6 | W X Y | 30 | W X Z | × | W Y Z | 12 | X Y Z | 24 |

5F+5I+5L | 5F+5I+5N | 5F+5I+5P | 5F+5I+5T | 5F+5I+5U |
---|---|---|---|---|

5F+5I+5V | 5F+5I+5W | 5F+5I+5X | 5F+5I+5Y | 5F+5I+5Z |

5F+5L+5N | 5F+5L+5P | 5F+5L+5T | 5F+5L+5U | 5F+5L+5V |

5F+5L+5W | 5F+5L+5X | 5F+5L+5Y | 5F+5L+5Z | 5F+5N+5P |

5F+5N+5T | 5F+5N+5U | 5F+5N+5V | 5F+5N+5W | 5F+5N+5X |

5F+5N+5Y | 5F+5N+5Z | 5F+5P+5T | 5F+5P+5U | 5F+5P+5V |

5F+5P+5W | 5F+5P+5X | 5F+5P+5Y | 5F+5P+5Z | 5F+5T+5U |

5F+5T+5V | 5F+5T+5W | 5F+5T+5X | 5F+5T+5Y | 5F+5T+5Z |

5F+5U+5V | 5F+5U+5W | 5F+5U+5X | 5F+5U+5Y | 5F+5U+5Z |

5F+5V+5W | 5F+5V+5X | 5F+5V+5Y | 5F+5V+5Z | 5F+5W+5X |

5F+5W+5Y | 5F+5W+5Z | 5F+5X+5Y | 5F+5X+5Z | 5F+5Y+5Z |

5I+5L+5N | 5I+5L+5P | 5I+5L+5T | 5I+5L+5U | 5I+5L+5V |

5I+5L+5W | 5I+5L+5X | 5I+5L+5Y | 5I+5L+5Z | 5I+5N+5P |

5I+5N+5T | 5I+5N+5U | 5I+5N+5V | 5I+5N+5W | 5I+5N+5X |

5I+5N+5Y | 5I+5N+5Z | 5I+5P+5T | 5I+5P+5U | 5I+5P+5V |

5I+5P+5W | 5I+5P+5X | 5I+5P+5Y | 5I+5P+5Z | 5I+5T+5U |

5I+5T+5V | 5I+5T+5W | 5I+5T+5X | 5I+5T+5Y | 5I+5T+5Z |

5I+5U+5V | 5I+5U+5W | 5I+5U+5X | 5I+5U+5Y | 5I+5U+5Z |

5I+5V+5W | 5I+5V+5X | 5I+5V+5Y | 5I+5V+5Z | 5I+5W+5X |

5I+5W+5Y | 5I+5W+5Z | 5I+5X+5Y | 5I+5X+5Z | 5I+5Y+5Z |

5L+5N+5P | 5L+5N+5T | 5L+5N+5U | 5L+5N+5V | 5L+5N+5W |

5L+5N+5X | 5L+5N+5Y | 5L+5N+5Z | 5L+5P+5T | 5L+5P+5U |

5L+5P+5V | 5L+5P+5W | 5L+5P+5X | 5L+5P+5Y | 5L+5P+5Z |

5L+5T+5U | 5L+5T+5V | 5L+5T+5W | 5L+5T+5X | 5L+5T+5Y |

5L+5T+5Z | 5L+5U+5V | 5L+5U+5W | 5L+5U+5X | 5L+5U+5Y |

5L+5U+5Z | 5L+5V+5W | 5L+5V+5X | 5L+5V+5Y | 5L+5V+5Z |

5L+5W+5X | 5L+5W+5Y | 5L+5W+5Z | 5L+5X+5Y | 5L+5X+5Z |

5L+5Y+5Z | 5N+5P+5T | 5N+5P+5U | 5N+5P+5V | 5N+5P+5W |

5N+5P+5X | 5N+5P+5Y | 5N+5P+5Z | 5N+5T+5U | 5N+5T+5V |

5N+5T+5W | 5N+5T+5X | 5N+5T+5Y | 5N+5T+5Z | 5N+5U+5V |

5N+5U+5W | 5N+5U+5X | 5N+5U+5Y | 5N+5U+5Z | 5N+5V+5W |

5N+5V+5X | 5N+5V+5Y | 5N+5V+5Z | 5N+5W+5X | 5N+5W+5Y |

5N+5W+5Z | 5N+5X+5Y | 5N+5X+5Z | 5N+5Y+5Z | 5P+5T+5U |

5P+5T+5V | 5P+5T+5W | 5P+5T+5X | 5P+5T+5Y | 5P+5T+5Z |

5P+5U+5V | 5P+5U+5W | 5P+5U+5X | 5P+5U+5Y | 5P+5U+5Z |

5P+5V+5W | 5P+5V+5X | 5P+5V+5Y | 5P+5V+5Z | 5P+5W+5X |

5P+5W+5Y | 5P+5W+5Z | 5P+5X+5Y | 5P+5X+5Z | 5P+5Y+5Z |

5T+5U+5V | 5T+5U+5W | 5T+5U+5X | 5T+5U+5Y | 5T+5U+5Z |

5T+5V+5W | 5T+5V+5X | 5T+5V+5Y | 5T+5V+5Z | 5T+5W+5X |

5T+5W+5Y | 5T+5W+5Z | 5T+5X+5Y | 5T+5X+5Z | 5T+5Y+5Z |

5U+5V+5W | 5U+5V+5X | 5U+5V+5Y | 5U+5V+5Z | 5U+5W+5X |

5U+5W+5Y | 5U+5W+5Z | 5U+5X+5Y | 5U+5X+5Z | 5U+5Y+5Z |

5V+5W+5X | 5V+5W+5Y | 5V+5W+5Z | 5V+5X+5Y | 5V+5X+5Z |

5V+5Y+5Z | 5W+5X+5Y | 5W+5X+5Z | 5W+5Y+5Z | 5X+5Y+5Z |

Last revised 2021-02-28.

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Col. George Sicherman [ HOME | MAIL ]