# Tiling Right Trapezoidal Polyominoes with Three Pentominoes

• 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.

Here I study the problem of tiling a polyomino shaped like a right trapezoid with copies of three pentominoes, using at least one of each. Such a polyomino has three straight sides, two of them parallel, and one zigzag side. For this problem, the polyomino may be triangular.

If you find a smaller solution than one of mine or solve an unsolved case, please write!

## Nomenclature

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

## Table

This table shows the smallest total number of three pentominoes known to be able to tile a trapezoidal polyomino:

 FIL — FNV 9 FUZ — INU 10 IUY 6 LPV 5 LWZ 8 NVW 3 PVX 7 TXZ — FIN 9 FNW — FVW 12 INV 6 IUZ — LPW 3 LXY 7 NVX 20 PVY 5 TYZ — FIP 5 FNX — FVX — INW 12 IVW 11 LPX 6 LXZ — NVY 6 PVZ 5 UVW 27 FIT — FNY 6 FVY 10 INX 27 IVX — LPY 3 LYZ 7 NVZ 6 PWX 6 UVX — FIU — FNZ — FVZ — INY 6 IVY 3 LPZ 6 NPT 5 NWX — PWY 3 UVY 10 FIV — FPT 5 FWX — INZ 18 IVZ — LTU — NPU 5 NWY 4 PWZ 3 UVZ — FIW 12 FPU 3 FWY 6 IPT 5 IWX 21 LTV — NPV 5 NWZ — PXY 5 UWX — FIX — FPV 4 FWZ — IPU 5 IWY 4 LTW 6 NPW 5 NXY 10 PXZ 7 UWY 3 FIY 8 FPW 5 FXY 15 IPV 5 IWZ 14 LTX — NPX 7 NXZ — PYZ 3 UWZ — FIZ — FPX 7 FXZ — IPW 4 IXY 18 LTY 6 NPY 3 NYZ 8 TUV — UXY 15 FLN 6 FPY 4 FYZ 8 IPX 5 IXZ — LTZ — NPZ 5 PTU 6 TUW 15 UXZ — FLP 4 FPZ 5 ILN 3 IPY 3 IYZ 14 LUV — NTU 9 PTV 6 TUX — UYZ 9 FLT — FTU — ILP 5 IPZ 5 LNP 3 LUW 4 NTV 9 PTW 3 TUY 10 VWX 35 FLU — FTV — ILT — ITU — LNT 7 LUX — NTW 6 PTX 7 TUZ — VWY 5 FLV — FTW 15 ILU — ITV — LNU 6 LUY 6 NTX 33 PTY 5 TVW 10 VWZ 11 FLW 4 FTX — ILV — ITW 7 LNV 6 LUZ — NTY 6 PTZ 6 TVX — VXY 27 FLX — FTY 6 ILW 3 ITX — LNW 4 LVW 6 NTZ 20 PUV 6 TVY 15 VXZ — FLY 6 FTZ — ILX — ITY 10 LNX 10 LVX — NUV 9 PUW 5 TVZ — VYZ 3 FLZ — FUV — ILY 3 ITZ — LNY 3 LVY 6 NUW 14 PUX 6 TWX 18 WXY 7 FNP 5 FUW 3 ILZ — IUV — LNZ 6 LVZ — NUX — PUY 3 TWY 3 WXZ — FNT 15 FUX — INP 5 IUW 18 LPT 5 LWX 7 NUY 6 PUZ 5 TWZ 11 WYZ 4 FNU — FUY 6 INT 3 IUX — LPU 6 LWY 6 NUZ — PVW 3 TXY — XYZ —

## Solutions

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

## 35 Tiles

Last revised 2024-01-01.

Back to Polyomino and Polyking Tiling < Polyform Tiling < Polyform Curiosities
Col. George Sicherman [ HOME | MAIL ]