Three-Pentomino Holey Balanced Rectangles

Introduction

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.

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.

Rodolfo 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.

In Two-Pentomino Balanced Rectangles I study the related problem of tiling some rectangle with two pentominoes, using the same number of copies of each. Here I study the same problem, allowing one-cell holes in the rectangle. The holes must lie in the interior of the rectangle and must not touch one another.

Thanks to Jenard Cabilao for suggesting allowing holes in balanced pentomino rectangles.

Nomenclature

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

Table

This table shows the area of the smallest rectangle known to be tiled by three pentominoes, using at least one of each, with optional one-cell holes.

FIL	30	FUZ	33	IUY	30	LWZ	63	PVX	96
FIN	63	FVW	63	IUZ	48	LXY	32	PVY	16
FIP	45	FVX	156	IVW	32	LXZ	98	PVZ	30
FIT	48	FVY	32	IVX	102	LYZ	30	PWX	108
FIU	30	FVZ	60	IVY	30	NPT	30	PWY	30
FIV	30	FWX	×	IVZ	30	NPU	15	PWZ	30
FIW	63	FWY	60	IWX	130	NPV	30	PXY	48
FIX	64	FWZ	×	IWY	30	NPW	60	PXZ	99
FIY	48	FXY	96	IWZ	63	NPX	64	PYZ	30
FIZ	63	FXZ	×	IXY	66	NPY	30	TUV	32
FLN	30	FYZ	60	IXZ	117	NPZ	30	TUW	84
FLP	30	ILN	30	IYZ	32	NTU	48	TUX	81
FLT	30	ILP	30	LNP	30	NTV	48	TUY	30
FLU	30	ILT	32	LNT	32	NTW	60	TUZ	64
FLV	16	ILU	30	LNU	30	NTX	126	TVW	32
FLW	30	ILV	30	LNV	15	NTY	30	TVX	130
FLX	81	ILW	30	LNW	30	NTZ	64	TVY	16
FLY	32	ILX	63	LNX	63	NUV	30	TVZ	32
FLZ	60	ILY	30	LNY	30	NUW	63	TWX	187
FNP	60	ILZ	48	LNZ	30	NUX	64	TWY	30
FNT	64	INP	30	LPT	32	NUY	30	TWZ	132
FNU	30	INT	30	LPU	16	NUZ	30	TXY	64
FNV	30	INU	30	LPV	15	NVW	30	TXZ	×
FNW	×	INV	30	LPW	30	NVX	64	TYZ	60
FNX	×	INW	63	LPX	48	NVY	32	UVW	35
FNY	60	INX	64	LPY	30	NVZ	30	UVX	117
FNZ	×	INY	30	LPZ	30	NWX	×	UVY	33
FPT	60	INZ	60	LTU	32	NWY	60	UVZ	33
FPU	15	IPT	30	LTV	30	NWZ	×	UWX	336
FPV	30	IPU	30	LTW	49	NXY	63	UWY	30
FPW	60	IPV	30	LTX	30	NXZ	×	UWZ	96
FPX	132	IPW	32	LTY	15	NYZ	60	UXY	30
FPY	30	IPX	64	LTZ	63	PTU	30	UXZ	162
FPZ	78	IPY	30	LUV	16	PTV	30	UYZ	33
FTU	33	IPZ	30	LUW	30	PTW	30	VWX	132
FTV	32	ITU	32	LUX	63	PTX	60	VWY	33
FTW	120	ITV	48	LUY	30	PTY	30	VWZ	33
FTX	×	ITW	48	LUZ	33	PTZ	33	VXY	66
FTY	30	ITX	78	LVW	16	PUV	15	VXZ	132
FTZ	×	ITY	30	LVX	64	PUW	48	VYZ	30
FUV	48	ITZ	63	LVY	32	PUX	30	WXY	96
FUW	30	IUV	32	LVZ	16	PUY	15	WXZ	×
FUX	156	IUW	63	LWX	63	PUZ	30	WYZ	60
FUY	33	IUX	80	LWY	30	PVW	30	XYZ	99

Solutions

So far as I know, these solutions have minimal area. They are not necessarily uniquely minimal. For a given area, the number of holes is minimal.

Area 15

Area 16

Area 30

Area 32

Area 33

Area 45

Area 48

Area 49

Area 60

Area 63

Area 64

Area 66

Area 78

Area 80

Area 81

Area 84

Area 96

Area 98

Area 99

Area 102

Area 108

Area 117

Area 120

Area 126

Area 130

Area 132

Area 156

Area 162

Area 187

Area 336

Last revised 2022-05-22.

Back to Polyform Tiling < Polyform Curiosities

Col. George Sicherman [ HOME | MAIL ]

FIL	30	FUZ	33	IUY	30	LWZ	63	PVX	96
FIN	63	FVW	63	IUZ	48	LXY	32	PVY	16
FIP	45	FVX	156	IVW	32	LXZ	98	PVZ	30
FIT	48	FVY	32	IVX	102	LYZ	30	PWX	108
FIU	30	FVZ	60	IVY	30	NPT	30	PWY	30
FIV	30	FWX	×	IVZ	30	NPU	15	PWZ	30
FIW	63	FWY	60	IWX	130	NPV	30	PXY	48
FIX	64	FWZ	×	IWY	30	NPW	60	PXZ	99
FIY	48	FXY	96	IWZ	63	NPX	64	PYZ	30
FIZ	63	FXZ	×	IXY	66	NPY	30	TUV	32
FLN	30	FYZ	60	IXZ	117	NPZ	30	TUW	84
FLP	30	ILN	30	IYZ	32	NTU	48	TUX	81
FLT	30	ILP	30	LNP	30	NTV	48	TUY	30
FLU	30	ILT	32	LNT	32	NTW	60	TUZ	64
FLV	16	ILU	30	LNU	30	NTX	126	TVW	32
FLW	30	ILV	30	LNV	15	NTY	30	TVX	130
FLX	81	ILW	30	LNW	30	NTZ	64	TVY	16
FLY	32	ILX	63	LNX	63	NUV	30	TVZ	32
FLZ	60	ILY	30	LNY	30	NUW	63	TWX	187
FNP	60	ILZ	48	LNZ	30	NUX	64	TWY	30
FNT	64	INP	30	LPT	32	NUY	30	TWZ	132
FNU	30	INT	30	LPU	16	NUZ	30	TXY	64
FNV	30	INU	30	LPV	15	NVW	30	TXZ	×
FNW	×	INV	30	LPW	30	NVX	64	TYZ	60
FNX	×	INW	63	LPX	48	NVY	32	UVW	35
FNY	60	INX	64	LPY	30	NVZ	30	UVX	117
FNZ	×	INY	30	LPZ	30	NWX	×	UVY	33
FPT	60	INZ	60	LTU	32	NWY	60	UVZ	33
FPU	15	IPT	30	LTV	30	NWZ	×	UWX	336
FPV	30	IPU	30	LTW	49	NXY	63	UWY	30
FPW	60	IPV	30	LTX	30	NXZ	×	UWZ	96
FPX	132	IPW	32	LTY	15	NYZ	60	UXY	30
FPY	30	IPX	64	LTZ	63	PTU	30	UXZ	162
FPZ	78	IPY	30	LUV	16	PTV	30	UYZ	33
FTU	33	IPZ	30	LUW	30	PTW	30	VWX	132
FTV	32	ITU	32	LUX	63	PTX	60	VWY	33
FTW	120	ITV	48	LUY	30	PTY	30	VWZ	33
FTX	×	ITW	48	LUZ	33	PTZ	33	VXY	66
FTY	30	ITX	78	LVW	16	PUV	15	VXZ	132
FTZ	×	ITY	30	LVX	64	PUW	48	VYZ	30
FUV	48	ITZ	63	LVY	32	PUX	30	WXY	96
FUW	30	IUV	32	LVZ	16	PUY	15	WXZ	×
FUX	156	IUW	63	LWX	63	PUZ	30	WYZ	60
FUY	33	IUX	80	LWY	30	PVW	30	XYZ	99

FIL	30	FUZ	33	IUY	30	LWZ	63	PVX	96
FIN	63	FVW	63	IUZ	48	LXY	32	PVY	16
FIP	45	FVX	156	IVW	32	LXZ	98	PVZ	30
FIT	48	FVY	32	IVX	102	LYZ	30	PWX	108
FIU	30	FVZ	60	IVY	30	NPT	30	PWY	30
FIV	30	FWX	×	IVZ	30	NPU	15	PWZ	30
FIW	63	FWY	60	IWX	130	NPV	30	PXY	48
FIX	64	FWZ	×	IWY	30	NPW	60	PXZ	99
FIY	48	FXY	96	IWZ	63	NPX	64	PYZ	30
FIZ	63	FXZ	×	IXY	66	NPY	30	TUV	32
FLN	30	FYZ	60	IXZ	117	NPZ	30	TUW	84
FLP	30	ILN	30	IYZ	32	NTU	48	TUX	81
FLT	30	ILP	30	LNP	30	NTV	48	TUY	30
FLU	30	ILT	32	LNT	32	NTW	60	TUZ	64
FLV	16	ILU	30	LNU	30	NTX	126	TVW	32
FLW	30	ILV	30	LNV	15	NTY	30	TVX	130
FLX	81	ILW	30	LNW	30	NTZ	64	TVY	16
FLY	32	ILX	63	LNX	63	NUV	30	TVZ	32
FLZ	60	ILY	30	LNY	30	NUW	63	TWX	187
FNP	60	ILZ	48	LNZ	30	NUX	64	TWY	30
FNT	64	INP	30	LPT	32	NUY	30	TWZ	132
FNU	30	INT	30	LPU	16	NUZ	30	TXY	64
FNV	30	INU	30	LPV	15	NVW	30	TXZ	×
FNW	×	INV	30	LPW	30	NVX	64	TYZ	60
FNX	×	INW	63	LPX	48	NVY	32	UVW	35
FNY	60	INX	64	LPY	30	NVZ	30	UVX	117
FNZ	×	INY	30	LPZ	30	NWX	×	UVY	33
FPT	60	INZ	60	LTU	32	NWY	60	UVZ	33
FPU	15	IPT	30	LTV	30	NWZ	×	UWX	336
FPV	30	IPU	30	LTW	49	NXY	63	UWY	30
FPW	60	IPV	30	LTX	30	NXZ	×	UWZ	96
FPX	132	IPW	32	LTY	15	NYZ	60	UXY	30
FPY	30	IPX	64	LTZ	63	PTU	30	UXZ	162
FPZ	78	IPY	30	LUV	16	PTV	30	UYZ	33
FTU	33	IPZ	30	LUW	30	PTW	30	VWX	132
FTV	32	ITU	32	LUX	63	PTX	60	VWY	33
FTW	120	ITV	48	LUY	30	PTY	30	VWZ	33
FTX	×	ITW	48	LUZ	33	PTZ	33	VXY	66
FTY	30	ITX	78	LVW	16	PUV	15	VXZ	132
FTZ	×	ITY	30	LVX	64	PUW	48	VYZ	30
FUV	48	ITZ	63	LVY	32	PUX	30	WXY	96
FUW	30	IUV	32	LVZ	16	PUY	15	WXZ	×
FUX	156	IUW	63	LWX	63	PUZ	30	WYZ	60
FUY	33	IUX	80	LWY	30	PVW	30	XYZ	99

FIL	30	FUZ	33	IUY	30	LWZ	63	PVX	96
FIN	63	FVW	63	IUZ	48	LXY	32	PVY	16
FIP	45	FVX	156	IVW	32	LXZ	98	PVZ	30
FIT	48	FVY	32	IVX	102	LYZ	30	PWX	108
FIU	30	FVZ	60	IVY	30	NPT	30	PWY	30
FIV	30	FWX	×	IVZ	30	NPU	15	PWZ	30
FIW	63	FWY	60	IWX	130	NPV	30	PXY	48
FIX	64	FWZ	×	IWY	30	NPW	60	PXZ	99
FIY	48	FXY	96	IWZ	63	NPX	64	PYZ	30
FIZ	63	FXZ	×	IXY	66	NPY	30	TUV	32
FLN	30	FYZ	60	IXZ	117	NPZ	30	TUW	84
FLP	30	ILN	30	IYZ	32	NTU	48	TUX	81
FLT	30	ILP	30	LNP	30	NTV	48	TUY	30
FLU	30	ILT	32	LNT	32	NTW	60	TUZ	64
FLV	16	ILU	30	LNU	30	NTX	126	TVW	32
FLW	30	ILV	30	LNV	15	NTY	30	TVX	130
FLX	81	ILW	30	LNW	30	NTZ	64	TVY	16
FLY	32	ILX	63	LNX	63	NUV	30	TVZ	32
FLZ	60	ILY	30	LNY	30	NUW	63	TWX	187
FNP	60	ILZ	48	LNZ	30	NUX	64	TWY	30
FNT	64	INP	30	LPT	32	NUY	30	TWZ	132
FNU	30	INT	30	LPU	16	NUZ	30	TXY	64
FNV	30	INU	30	LPV	15	NVW	30	TXZ	×
FNW	×	INV	30	LPW	30	NVX	64	TYZ	60
FNX	×	INW	63	LPX	48	NVY	32	UVW	35
FNY	60	INX	64	LPY	30	NVZ	30	UVX	117
FNZ	×	INY	30	LPZ	30	NWX	×	UVY	33
FPT	60	INZ	60	LTU	32	NWY	60	UVZ	33
FPU	15	IPT	30	LTV	30	NWZ	×	UWX	336
FPV	30	IPU	30	LTW	49	NXY	63	UWY	30
FPW	60	IPV	30	LTX	30	NXZ	×	UWZ	96
FPX	132	IPW	32	LTY	15	NYZ	60	UXY	30
FPY	30	IPX	64	LTZ	63	PTU	30	UXZ	162
FPZ	78	IPY	30	LUV	16	PTV	30	UYZ	33
FTU	33	IPZ	30	LUW	30	PTW	30	VWX	132
FTV	32	ITU	32	LUX	63	PTX	60	VWY	33
FTW	120	ITV	48	LUY	30	PTY	30	VWZ	33
FTX	×	ITW	48	LUZ	33	PTZ	33	VXY	66
FTY	30	ITX	78	LVW	16	PUV	15	VXZ	132
FTZ	×	ITY	30	LVX	64	PUW	48	VYZ	30
FUV	48	ITZ	63	LVY	32	PUX	30	WXY	96
FUW	30	IUV	32	LVZ	16	PUY	15	WXZ	×
FUX	156	IUW	63	LWX	63	PUZ	30	WYZ	60
FUY	33	IUX	80	LWY	30	PVW	30	XYZ	99