# Scaled Three-Pentomino L Shapes

• 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.

Only five scaled pentominoes can tile L-shaped polyominoes:

Here I show which sets of three pentominoes can tile an L-shaped polyomino, using the pentominoes at various sizes. If you find a solution with fewer tiles than one of mine, please write!

Bryce Herdt contributed improvements.

## Nomenclature

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

## Table

This table shows the fewest scaled pentominoes known to be able to tile some L-shaped polyomino, using at least one of each pentomino. The L-shaped polyominoes are not necessarily the smallest that can be tiled, only the smallest that can be tiled with the fewest tiles.

 FIL 4 FNV 6 FUZ 5 INU 5 IUY 3 LPV 3 LWZ 6 NVW 8 PVX 6 TXZ — FIN 7 FNW — FVW 10 INV 4 IUZ 7 LPW 4 LXY 6 NVX 14 PVY 3 TYZ 6 FIP 4 FNX — FVX 7 INW 7 IVW 5 LPX 6 LXZ 8 NVY 5 PVZ 3 UVW 13 FIT 9 FNY 5 FVY 5 INX 13 IVX 10 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 9 FPU 3 FWY 6 IPT 3 IWX 13 LTV 4 NPV 3 NWZ — PXY 5 UWX 14 FIX 14 FPV 4 FWZ — IPU 3 IWY 5 LTW 5 NPW 5 NXY 6 PXZ 6 UWY 4 FIY 5 FPW 5 FXY 7 IPV 3 IWZ 7 LTX 8 NPX 6 NXZ — PYZ 4 UWZ 14 FIZ 7 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 14 LTZ 5 NPZ 4 PTU 4 TUW 15 UXZ 11 FLP 3 FPZ 4 ILN 3 IPY 3 IYZ 6 LUV 5 NTU 4 PTV 3 TUX 12 UYZ 5 FLT 6 FTU 5 ILP 3 IPZ 3 LNP 3 LUW 6 NTV 5 PTW 4 TUY 5 VWX 27 FLU 3 FTV 10 ILT 3 ITU 7 LNT 4 LUX 3 NTW 11 PTX 6 TUZ 7 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 17 VXY 6 FLX 8 FTY 6 ILW 4 ITX 11 LNW 6 LVW 5 NTZ 11 PUV 4 TVY 5 VXZ 10 FLY 5 FTZ — ILX 6 ITY 5 LNX 6 LVX 6 NUV 6 PUW 5 TVZ 6 VYZ 4 FLZ 4 FUV 3 ILY 3 ITZ 9 LNY 4 LVY 4 NUW 7 PUX 3 TWX 17 WXY 7 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 9 LPT 3 LWX 7 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 7 XYZ 7

## Solutions

So far as I know, these solutions have the fewest possible tiles. They are not necessarily uniquely minimal.

### 27 Tiles

Last revised 2023-03-18.

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Col. George Sicherman [ HOME | MAIL ]