# Two-Pentomino Square Frames

A *pentomino* is a figure made of five squares joined
edge to edge.
There are 12 such figures, not distinguishing reflections and rotations.
They were first enumerated and studied by Solomon Golomb.
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 two pentominoes.
Thanks to Joyce Michel for suggesting this problem.

Bryce Herdt solved a balanced variant.

I use Solomon W. Golomb's original names for the pentominoes:

| F | I | L | N | P | T | U | V | W | X | Y | Z |

F | • | 28 | 8 | — | 12 | — | 16 | 12 | — | — | 8 | — |

I | 28 | • | 8 | 12 | 8 | 8 | 24 | 8 | 28 | ? | 8 | 64 |

L | 8 | 8 | • | 8 | 8 | 12 | 12 | 8 | 12 | 28 | 8 | 16 |

N | — | 12 | 8 | • | 8 | 8 | 16 | 12 | — | — | 8 | — |

P | 12 | 8 | 8 | 8 | • | 12 | 8 | 12 | 12 | 12 | 8 | 12 |

T | — | 8 | 12 | 8 | 12 | • | 12 | — | 12 | — | 16 | — |

U | 16 | 24 | 12 | 16 | 8 | 12 | • | 88 | — | 12 | 12 | — |

V | 12 | 8 | 8 | 12 | 12 | — | 88 | • | 64 | — | 8 | 8 |

W | — | 28 | 12 | — | 12 | 12 | — | 64 | • | — | 12 | — |

X | — | ? | 28 | — | 12 | — | 12 | — | — | • | 24 | — |

Y | 8 | 8 | 8 | 8 | 8 | 16 | 12 | 8 | 12 | 24 | • | 16 |

Z | — | 64 | 16 | — | 12 | — | — | 8 | — | — | 16 | • |

So far as I know, these solutions
have minimal area. They are not necessarily uniquely minimal.
### 8 Tiles

### 12 Tiles

### 16 Tiles

### 24 Tiles

### 28 Tiles

### 64 Tiles

### 88 Tiles

A balanced tiling has equal numbers of the two pentominoes.
Here I do not show tilings from the previous section
if they are already balanced.
### 12 Tiles

### 16 Tiles

### 24 Tiles

### 32 Tiles

### 36 Tiles

### 40 Tiles

### 64 Tiles

### 96 Tiles

### 120 Tiles

### 544 Tiles

In a contiguous variant, all the pentominoes with a given shape
are connected at edges.
Here I do not show tilings from the first section
if they are already contiguous.
### 8 Tiles

### 12 Tiles

### 16 Tiles

### 24 Tiles

### 32 Tiles

### 40 Tiles

### 48 Tiles

### 72 Tiles

Last revised 2023-08-03.

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Polyform Curiosities

Col. George Sicherman
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