The January 2008 issue of Erich Friedman's Math Magic defined a frame as a square polyomino with a centered square hole. The problem was to find the frame with least area that could be tiled with a given polyomino.
Here I study the related problem of finding the smallest frame that can be tiled with copies of three pentominoes.
See also Two-Pentomino Square Frames.
F I L | 12 | F I N | 12 | F I P | 12 | F I T | 16 | F I U | 12 | F I V | 8 | F I W | 12 | F I X | 28 | F I Y | 8 | F I Z | 24 |
F L N | 8 | F L P | 12 | F L T | 12 | F L U | 12 | F L V | 12 | F L W | 12 | F L X | 16 | F L Y | 12 | F L Z | 16 | F N P | 8 |
F N T | 16 | F N U | 12 | F N V | 8 | F N W | ? | F N X | ? | F N Y | 12 | F N Z | ? | F P T | 12 | F P U | 12 | F P V | 12 |
F P W | 12 | F P X | 16 | F P Y | 8 | F P Z | 12 | F T U | 16 | F T V | 16 | F T W | 16 | F T X | ? | F T Y | 12 | F T Z | ? |
F U V | 12 | F U W | 16 | F U X | 16 | F U Y | 12 | F U Z | 12 | F V W | 24 | F V X | 24 | F V Y | 12 | F V Z | 16 | F W X | ? |
F W Y | 12 | F W Z | ? | F X Y | 16 | F X Z | ? | F Y Z | 12 | I L N | 8 | I L P | 8 | I L T | 12 | I L U | 12 | I L V | 8 |
I L W | 12 | I L X | 16 | I L Y | 8 | I L Z | 12 | I N P | 8 | I N T | 12 | I N U | 16 | I N V | 8 | I N W | 16 | I N X | 28 |
I N Y | 8 | I N Z | 16 | I P T | 12 | I P U | 8 | I P V | 12 | I P W | 12 | I P X | 12 | I P Y | 8 | I P Z | 12 | I T U | 16 |
I T V | 12 | I T W | 16 | I T X | 28 | I T Y | 12 | I T Z | 16 | I U V | 16 | I U W | 16 | I U X | 12 | I U Y | 12 | I U Z | 16 |
I V W | 24 | I V X | 40 | I V Y | 12 | I V Z | 12 | I W X | 56 | I W Y | 12 | I W Z | 16 | I X Y | 24 | I X Z | 28 | I Y Z | 12 |
L N P | 8 | L N T | 12 | L N U | 12 | L N V | 8 | L N W | 12 | L N X | 16 | L N Y | 8 | L N Z | 12 | L P T | 12 | L P U | 8 |
L P V | 8 | L P W | 12 | L P X | 12 | L P Y | 12 | L P Z | 12 | L T U | 12 | L T V | 12 | L T W | 16 | L T X | 24 | L T Y | 8 |
L T Z | 16 | L U V | 12 | L U W | 12 | L U X | 12 | L U Y | 12 | L U Z | 12 | L V W | 8 | L V X | 16 | L V Y | 8 | L V Z | 8 |
L W X | 24 | L W Y | 12 | L W Z | 16 | L X Y | 16 | L X Z | 16 | L Y Z | 12 | N P T | 8 | N P U | 8 | N P V | 8 | N P W | 12 |
N P X | 16 | N P Y | 8 | N P Z | 12 | N T U | 12 | N T V | 12 | N T W | 12 | N T X | 24 | N T Y | 12 | N T Z | 16 | N U V | 12 |
N U W | 12 | N U X | 16 | N U Y | 12 | N U Z | 16 | N V W | 12 | N V X | 16 | N V Y | 12 | N V Z | 12 | N W X | ? | N W Y | 12 |
N W Z | ? | N X Y | 12 | N X Z | ? | N Y Z | 12 | P T U | 12 | P T V | 12 | P T W | 12 | P T X | 12 | P T Y | 8 | P T Z | 12 |
P U V | 12 | P U W | 12 | P U X | 12 | P U Y | 8 | P U Z | 12 | P V W | 12 | P V X | 12 | P V Y | 8 | P V Z | 8 | P W X | 16 |
P W Y | 12 | P W Z | 12 | P X Y | 12 | P X Z | 12 | P Y Z | 12 | T U V | 16 | T U W | 12 | T U X | 16 | T U Y | 16 | T U Z | 24 |
T V W | 16 | T V X | ? | T V Y | 12 | T V Z | 12 | T W X | 24 | T W Y | 16 | T W Z | 16 | T X Y | 16 | T X Z | ? | T Y Z | 16 |
U V W | 40 | U V X | 16 | U V Y | 12 | U V Z | 12 | U W X | 24 | U W Y | 12 | U W Z | ? | U X Y | 16 | U X Z | 48 | U Y Z | 12 |
V W X | 56 | V W Y | 12 | V W Z | 12 | V X Y | 12 | V X Z | 56 | V Y Z | 12 | W X Y | 16 | W X Z | ? | W Y Z | 12 | X Y Z | 16 |
Last revised 2023-08-06.