# Three-Pentomino 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.

The problem of arranging copies of a polyomino to form a rectangle has been studied for a long time. The problem of arranging copies of two polyominoes to form a rectangle was studied by Earl Kramer and Frits Göbel in Tiling Rectangles with Pairs of Pentominoes, Journal of Recreational Mathematics 16(3) (1983), 198–206. Later it was studied in Rodolfo Kurchan's Puzzle Fun, Issue 19, Tyling with Pairs of Polyominoes. Still later, the problem was studied further in Erich Friedman's Math Magic for August 2010.

Here I study the problem of arranging copies of three pentominoes to form a rectangle, using at least one of each.

## Nomenclature

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

## Table of Results

This table shows the smallest total number of pentominoes known to be able to tile a rectangle with copies of three given pentominoes, using at least one of each.

 FIL 4 FNV 6 FUZ 5 INU 5 IUY 3 LPV 3 LWZ 6 NVW 8 PVX 6 TXZ — FIN 11 FNW — FVW 10 INV 4 IUZ 7 LPW 4 LXY 6 NVX 14 PVY 3 TYZ 6 FIP 4 FNX — FVX 15 INW 9 IVW 7 LPX 6 LXZ 8 NVY 5 PVZ 3 UVW 30 FIT 10 FNY 5 FVY 5 INX 19 IVX 24 LPY 3 LYZ 5 NVZ 3 PWX 6 UVX 4 FIU 3 FNZ — FVZ 5 INY 5 IVY 4 LPZ 3 NPT 3 NWX — PWY 3 UVY 3 FIV 5 FPT 4 FWX — INZ 7 IVZ 3 LTU 5 NPU 3 NWY 7 PWZ 4 UVZ 5 FIW 11 FPU 3 FWY 6 IPT 3 IWX 20 LTV 4 NPV 3 NWZ — PXY 5 UWX 23 FIX 15 FPV 4 FWZ — IPU 3 IWY 5 LTW 5 NPW 5 NXY 7 PXZ 6 UWY 4 FIY 5 FPW 5 FXY 7 IPV 3 IWZ 7 LTX 8 NPX 6 NXZ — PYZ 4 UWZ — FIZ 15 FPX 8 FXZ — IPW 4 IXY 6 LTY 4 NPY 3 NYZ 6 TUV 5 UXY 5 FLN 5 FPY 4 FYZ 5 IPX 5 IXZ 15 LTZ 5 NPZ 4 PTU 4 TUW 15 UXZ 23 FLP 3 FPZ 4 ILN 3 IPY 3 IYZ 6 LUV 5 NTU 4 PTV 3 TUX 17 UYZ 5 FLT 6 FTU 5 ILP 3 IPZ 3 LNP 3 LUW 6 NTV 5 PTW 4 TUY 5 VWX 43 FLU 3 FTV 10 ILT 3 ITU 7 LNT 4 LUX 3 NTW 11 PTX 6 TUZ 16 VWY 5 FLV 4 FTW 14 ILU 4 ITV 6 LNU 4 LUY 3 NTX 11 PTY 3 TVW 5 VWZ 8 FLW 5 FTX — ILV 3 ITW 7 LNV 4 LUZ 5 NTY 5 PTZ 5 TVX — VXY 8 FLX 8 FTY 6 ILW 4 ITX 17 LNW 6 LVW 5 NTZ 11 PUV 4 TVY 5 VXZ 21 FLY 5 FTZ — ILX 6 ITY 5 LNX 6 LVX 8 NUV 6 PUW 5 TVZ 6 VYZ 4 FLZ 4 FUV 3 ILY 3 ITZ 10 LNY 4 LVY 4 NUW 7 PUX 3 TWX 20 WXY 9 FNP 4 FUW 4 ILZ 5 IUV 7 LNZ 5 LVZ 3 NUX 7 PUY 3 TWY 7 WXZ — FNT 10 FUX 5 INP 3 IUW 11 LPT 3 LWX 9 NUY 4 PUZ 4 TWZ 14 WYZ 6 FNU 4 FUY 3 INT 6 IUX 4 LPU 3 LWY 5 NUZ 8 PVW 4 TXY 9 XYZ 7

## Solutions

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

### 60 Tiles

Last revised 2023-03-08.

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Col. George Sicherman [ HOME | MAIL ]