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 |
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| 5F+5I+5V | 5F+5I+5W | 5F+5I+5X | 5F+5I+5Y | 5F+5I+5Z |
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| 5F+5L+5N | 5F+5L+5P | 5F+5L+5T | 5F+5L+5U | 5F+5L+5V |
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| 5F+5L+5W | 5F+5L+5X | 5F+5L+5Y | 5F+5L+5Z | 5F+5N+5P |
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| 5F+5N+5T | 5F+5N+5U | 5F+5N+5V | 5F+5N+5W | 5F+5N+5X |
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| 5F+5N+5Y | 5F+5N+5Z | 5F+5P+5T | 5F+5P+5U | 5F+5P+5V |
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| 5F+5P+5W | 5F+5P+5X | 5F+5P+5Y | 5F+5P+5Z | 5F+5T+5U |
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| 5F+5T+5V | 5F+5T+5W | 5F+5T+5X | 5F+5T+5Y | 5F+5T+5Z |
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| 5F+5U+5V | 5F+5U+5W | 5F+5U+5X | 5F+5U+5Y | 5F+5U+5Z |
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| 5F+5V+5W | 5F+5V+5X | 5F+5V+5Y | 5F+5V+5Z | 5F+5W+5X |
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| 5F+5W+5Y | 5F+5W+5Z | 5F+5X+5Y | 5F+5X+5Z | 5F+5Y+5Z |
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| 5I+5L+5N | 5I+5L+5P | 5I+5L+5T | 5I+5L+5U | 5I+5L+5V |
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| 5I+5L+5W | 5I+5L+5X | 5I+5L+5Y | 5I+5L+5Z | 5I+5N+5P |
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| 5I+5N+5T | 5I+5N+5U | 5I+5N+5V | 5I+5N+5W | 5I+5N+5X |
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| 5I+5N+5Y | 5I+5N+5Z | 5I+5P+5T | 5I+5P+5U | 5I+5P+5V |
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| 5I+5P+5W | 5I+5P+5X | 5I+5P+5Y | 5I+5P+5Z | 5I+5T+5U |
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| 5I+5T+5V | 5I+5T+5W | 5I+5T+5X | 5I+5T+5Y | 5I+5T+5Z |
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| 5I+5U+5V | 5I+5U+5W | 5I+5U+5X | 5I+5U+5Y | 5I+5U+5Z |
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| 5I+5V+5W | 5I+5V+5X | 5I+5V+5Y | 5I+5V+5Z | 5I+5W+5X |
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| 5I+5W+5Y | 5I+5W+5Z | 5I+5X+5Y | 5I+5X+5Z | 5I+5Y+5Z |
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| 5L+5N+5P | 5L+5N+5T | 5L+5N+5U | 5L+5N+5V | 5L+5N+5W |
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| 5L+5N+5X | 5L+5N+5Y | 5L+5N+5Z | 5L+5P+5T | 5L+5P+5U |
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| 5L+5P+5V | 5L+5P+5W | 5L+5P+5X | 5L+5P+5Y | 5L+5P+5Z |
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| 5L+5T+5U | 5L+5T+5V | 5L+5T+5W | 5L+5T+5X | 5L+5T+5Y |
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| 5L+5T+5Z | 5L+5U+5V | 5L+5U+5W | 5L+5U+5X | 5L+5U+5Y |
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| 5L+5U+5Z | 5L+5V+5W | 5L+5V+5X | 5L+5V+5Y | 5L+5V+5Z |
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| 5L+5W+5X | 5L+5W+5Y | 5L+5W+5Z | 5L+5X+5Y | 5L+5X+5Z |
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| 5L+5Y+5Z | 5N+5P+5T | 5N+5P+5U | 5N+5P+5V | 5N+5P+5W |
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| 5N+5P+5X | 5N+5P+5Y | 5N+5P+5Z | 5N+5T+5U | 5N+5T+5V |
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| 5N+5T+5W | 5N+5T+5X | 5N+5T+5Y | 5N+5T+5Z | 5N+5U+5V |
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| 5N+5U+5W | 5N+5U+5X | 5N+5U+5Y | 5N+5U+5Z | 5N+5V+5W |
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| 5N+5V+5X | 5N+5V+5Y | 5N+5V+5Z | 5N+5W+5X | 5N+5W+5Y |
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| 5N+5W+5Z | 5N+5X+5Y | 5N+5X+5Z | 5N+5Y+5Z | 5P+5T+5U |
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| 5P+5T+5V | 5P+5T+5W | 5P+5T+5X | 5P+5T+5Y | 5P+5T+5Z |
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| 5P+5U+5V | 5P+5U+5W | 5P+5U+5X | 5P+5U+5Y | 5P+5U+5Z |
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| 5P+5V+5W | 5P+5V+5X | 5P+5V+5Y | 5P+5V+5Z | 5P+5W+5X |
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| 5P+5W+5Y | 5P+5W+5Z | 5P+5X+5Y | 5P+5X+5Z | 5P+5Y+5Z |
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| 5T+5U+5V | 5T+5U+5W | 5T+5U+5X | 5T+5U+5Y | 5T+5U+5Z |
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| 5T+5V+5W | 5T+5V+5X | 5T+5V+5Y | 5T+5V+5Z | 5T+5W+5X |
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| 5T+5W+5Y | 5T+5W+5Z | 5T+5X+5Y | 5T+5X+5Z | 5T+5Y+5Z |
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| 5U+5V+5W | 5U+5V+5X | 5U+5V+5Y | 5U+5V+5Z | 5U+5W+5X |
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| 5U+5W+5Y | 5U+5W+5Z | 5U+5X+5Y | 5U+5X+5Z | 5U+5Y+5Z |
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| 5V+5W+5X | 5V+5W+5Y | 5V+5W+5Z | 5V+5X+5Y | 5V+5X+5Z |
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| 5V+5Y+5Z | 5W+5X+5Y | 5W+5X+5Z | 5W+5Y+5Z | 5X+5Y+5Z |
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Last revised 2021-02-28.