# Three-Pentomino Holey Balanced Rectangles

• Introduction
• Nomenclature
• Table
• Solutions
• ## 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 336

Last revised 2022-05-22.

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Col. George Sicherman [ HOME | MAIL ]