## Introduction

A polyomino is a plane figure made of equal squares joined edge to edge. Polyominoes were first enumerated and studied by Solomon Golomb.

The compatibility problem is to find a figure that can be tiled with each of a set of polyforms. Giovanni Resta's website Polypolyominoes (Polipolimini in Italian) presents compatibility results for polyominoes with as many as 10 cells. It is the Web's main resource for polyomino compatibility.

Since Resta assembled and posted his results, I have found a few new or smaller solutions. It is hard for Resta to add or change solutions in his compact web graphic format, so I present my solutions here instead. Resta shows some of my solutions on his main page and his individual pages.

I do not claim to have found all possible new and improved solutions. If you find others, please write!

## Solutions

Small polyominoes have conventional letter names. For example, the five tetrominoes are named I, L, N, T, and O (or Q in Italian). Resta uses strings of letters to identify polyminoes with more than five cells. Here I adopt Resta's names to aid in coordinating with his pages. Exclamation points (!) indicate solutions to formerly unsolved cases. Names for polyominoes of orders 1 through 8 appear on this page of Resta's.

• Hexominoes and Pentominoes
• Hexominoes and Hexominoes
• Heptominoes and Tetrominoes
• Heptominoes and Pentominoes FILN
• Heptominoes and Pentominoes PTUV
• Heptominoes and Pentominoes WXYZ
• Octominoes and Tromino I
• Octominoes and Tetromino I
• Octominoes and Tetromino N
• Octominoes and Tetromino Q
• Octominoes and Pentominoes
• Enneominoes and Tetromino I
• Enneominoes and Tetromino Q
• Decominoes and Tetromino L
• ### Hexominoes and Pentominoes

<Giovanni Resta's Hexominoes and Pentominoes>

6AC = 5Y6AE = 5U6BG = 5X!6BH = 5Z
5 / 6 40 / 48 120 / 144 40 / 48

### Hexominoes and Hexominoes

<Giovanni Resta's Hexominoes and Hexominoes>

6AX = 6AZ!
68 / 68

### Heptominoes and Tetrominoes

<Giovanni Resta's Heptominoes and Tetrominoes>

7BX = 4Q 7CY = 4N 7DY = 4T! 7EB = 4Q
16 / 28 32 / 56 32 / 56 32 / 56
7CU = 4Q!
128 / 224

### Heptominoes and Pentominoes FILN

<Giovanni Resta's Heptominoes and Pentominoes FILN>

7AP = 5N!7AV = 5L!7AY = 5I7BN = 5I
70 / 98 40 / 56 20 / 28 30 / 42

### Heptominoes and Pentominoes PTUV

<Giovanni Resta's Heptominoes and Pentominoes PTUV>

7AI = 5U 7AV = 5P! 7AY = 5U 7BL = 5T
30 / 42 40 / 56 30 / 42 30 / 42
7CD = 5U 7CW = 5V 7CX = 5V 7DA = 5V
30 / 42 30 / 42 20 / 28 20 / 28
7DF = 5T 7DG = 5T 7DN = 5T! 7DO = 5V
30 / 42 30 / 42 40 / 56 30 / 42
7DR = 5T 7DW = 5U 7DX = 5T
30 / 42 20 / 28 30 / 42

### Heptominoes and Pentominoes WXYZ

<Giovanni Resta's Heptominoes and Pentominoes WXYZ>

7AN = 5Z7AT = 5Z7BL = 5Z7CB = 5Z
30 / 42 20 / 28 30 / 42 20 / 28
7CH = 5W7CJ = 5Z7CQ = 5X!
30 / 42 30 / 42 100 / 140
7CV = 5Z 7DE = 5Z! 7DL = 5Y 7DQ = 5Z!
30 / 42 40 / 56 20 / 28 50 / 70

### Octominoes and Tromino I

<Giovanni Resta's Octominoes and Tromino I>

8KD = 3I8MY = 3I!
24 / 64 66 / 176

### Octominoes and Tetromino I

<Giovanni Resta's Octominoes and Tetromino I>

8DP = 4I
28 / 56

### Octominoes and Tetromino N

<Giovanni Resta's Octominoes and Tetromino N>

8LZ = 4N!
48 / 96

### Octominoes and Tetromino Q

<Giovanni Resta's Octominoes and Tetromino Q>

8BV = 4Q! 8DM = 4Q! 8JA = 4Q!
56 / 112 52 / 104 44 / 88
8KQ = 4Q! 8MO = 4Q 8MP = 4Q
28 / 56 28 / 56 24 / 48

### Octominoes and Pentominoes

<Giovanni Resta's Octominoes and Pentominoes>

8AF = 5F! 8AF = 5V! 8AI = 5Z! 8AK = 5T!
60 / 96 20 / 32 20 / 32 20 / 32
8AL = 5T! 8AN = 5T! 8AN = 5W! 8AN = 5Z!
20 / 32 40 / 64 20 / 32 20 / 32
8AQ = 5T! 8AQ = 5V! 8AR = 5T!
40 / 64 20 / 32 20 / 32
8AZ = 5L! 8AZ = 5Z! 8BD = 5P!
20 / 32 20 / 32 20 / 32
8BD = 5Z! 8BF = 5V! 8BH = 5V! 8BQ = 5V!
20 / 32 20 / 32 20 / 32 20 / 32
8BR = 5T! 8BR = 5V! 8BX = 5V! 8BZ = 5U! 8CA = 5Z!
20 / 32 20 / 32 20 / 32 20 / 32 20 / 32
8CB = 5T! 8CD = 5I! 8CE = 5W! 8CH = 5T!
40 / 64 40 / 64 20 / 32 20 / 32
8CI = 5V! 8CJ = 5T! 8CK = 5T! 8CN = 5Z! 8CO = 5T!
20 / 32 20 / 32 20 / 32 20 / 32 20 / 32
8CR = 5W! 8CU = 5U! 8CU = 5V!
40 / 64 40 / 64 20 / 32
8CU = 5W! 8CW = 5Z! 8CZ = 5T!
40 / 64 20 / 32 20 / 32
8DA = 5W! 8DE = 5N! 8DF = 5Z! 8DG = 5U!
40 / 64 10 / 16 20 / 32 20 / 32
8DH = 5U! 8DI = 5V! 8DL = 5U! 8DM = 5V!
20 / 32 40 / 64 20 / 32 40 / 64
8DO = 5T! 8DO = 5U! 8DQ = 5U!
40 / 64 20 / 32 40 / 64
8DR = 5T! 8DU = 5U! 8DV = 5V! 8DX = 5Z!
40 / 64 20 / 32 20 / 32 20 / 32
8DY = 5T! 8EB = 5U! 8EB = 5W! 8EC = 5U!
20 / 32 20 / 32 20 / 32 20 / 32
8EC = 5Z! 8ED = 5U! 8EF = 5U!
40 / 64 40 / 64 20 / 32
8EF = 5Z! 8EM = 5V! 8EM = 5Z!
20 / 32 20 / 32 40 / 64
8EN = 5N! 8EP = 5I!
40 / 64 20 / 32
8EQ = 5W! 8ES = 5W! 8ET = 5Z! 8EW = 5W!
60 / 96 20 / 32 20 / 32 30 / 48
8FA = 5U! 8FA = 5W! 8FC = 5Z! 8FD = 5V!
20 / 32 40 / 64 20 / 32 20 / 32
8FD = 5Z! 8FE = 5F! 8FE = 5N!
20 / 32 40 / 64 30 / 48
8FH = 5T! 8FK = 5I! 8FK = 5Z! 8FM = 5U!
20 / 32 10 / 16 20 / 32 20 / 32
8FS = 5V! 8FT = 5I! 8FT = 5U! 8FV = 5V!
20 / 32 20 / 32 40 / 64 20 / 32
8FZ = 5F! 8FZ = 5I! 8GA = 5X! 8GD = 5T!
20 / 32 20 / 32 20 / 32 40 / 64
8GD = 5V! 8GF = 5U! 8GG = 5W!
20 / 32 160 / 256 40 / 64
8GJ = 5T! 8GK = 5T! 8GN = 5V! 8GO = 5U!
40 / 64 20 / 32 40 / 64 20 / 32
8GR = 5W! 8GS = 5U! 8GX = 5T! 8GX = 5U!
60 / 96 40 / 64 20 / 32 20 / 32
8GX = 5W! 8HC = 5U! 8HF = 5V!
40 / 64 20 / 32 20 / 32
8HH = 5T! 8HH = 5U! 8HH = 5Z!
20 / 32 30 / 48 40 / 64
8HK = 5Z! 8HL = 5Z! 8HN = 5W! 8HO = 5U!
20 / 32 20 / 32 20 / 32 20 / 32
8HO = 5W! 8HO = 5Z! 8HQ = 5T! 8HQ = 5Z!
40 / 64 20 / 32 20 / 32 20 / 32
8HT = 5V! 8HV = 5T! 8HW = 5T! 8HX = 5T!
20 / 32 20 / 32 20 / 32 20 / 32
8HY = 5T! 8HZ = 5V! 8HZ = 5Z! 8IA = 5T!
60 / 96 20 / 32 20 / 32 20 / 32
8IB = 5W! 8IB = 5Z! 8IC = 5T! 8II = 5I!
20 / 32 40 / 64 20 / 32 20 / 32
8II = 5Z! 8IK = 5T! 8IK = 5Z! 8IL = 5Z!
20 / 32 30 / 48 20 / 32 40 / 64
8IO = 5W! 8IT = 5U! 8IW = 5W! 8IY = 5T!
20 / 32 20 / 32 80 / 128 20 / 32
8JA = 5U! 8JA = 5Z! 8JC = 5T!
40 / 64 20 / 32 40 / 64
8JE = 5W! 8JH = 5W! 8JI = 5U! 8JJ = 5T! 8JL = 5F!
20 / 32 20 / 32 20 / 32 20 / 32 20 / 32
8JO = 5T! 8JQ = 5W! 8JS = 5Z! 8JT = 5W! 8JU = 5Z!
40 / 64 40 / 64 30 / 48 40 / 64 20 / 32
8JW = 5L! 8JX = 5V! 8JX = 5Z! 8JZ = 5T! 8KA = 5F!
50 / 80 20 / 32 20 / 32 20 / 32 20 / 32
8KB = 5T! 8KD = 5N! 8KE = 5U! 8KF = 5X!
40 / 64 40 / 64 20 / 32 20 / 32
8KG = 5T! 8KG = 5U! 8KH = 5T! 8KI = 5Z! 8KK = 5V!
20 / 32 20 / 32 20 / 32 20 / 32 20 / 32
8KK = 5W! 8KO = 5V! 8KO = 5Z! 8KP = 5Z!
40 / 64 20 / 32 20 / 32 20 / 32
8KS = 5T! 8KS = 5Z! 8KT = 5Z! 8KX = 5I!
20 / 32 40 / 64 20 / 32 60 / 96
8KX = 5T! 8KZ = 5V! 8LA = 5V! 8LB = 5Z! 8LC = 5T!
20 / 32 20 / 32 20 / 32 40 / 64 20 / 32
8LE = 5U! 8LE = 5W! 8LF = 5W! 8LG = 5W!
20 / 32 30 / 48 40 / 64 20 / 32
8LK = 5U! 8LK = 5V! 8LL = 5I! 8LM = 5Y!
20 / 32 20 / 32 20 / 32 80 / 128
8LN = 5Z! 8LO = 5T! 8LP = 5W! 8LP = 5Z!
20 / 32 40 / 64 20 / 32 10 / 16
8LQ = 5W! 8LS = 5W! 8LU = 5W!
20 / 32 80 / 128 60 / 96
8LV = 5U! 8LW = 5W! 8LX = 5U! 8LX = 5Z!
20 / 32 20 / 32 40 / 64 20 / 32
8LY = 5U! 8LZ = 5F! 8MC = 5T! 8MC = 5U! 8MC = 5Z!
20 / 32 20 / 32 20 / 32 20 / 32 40 / 64
8MD = 5U! 8MD = 5V! 8ME = 5U!
20 / 32 60 / 96 20 / 32
8MF = 5N! 8MI = 5U!
80 / 128 40 / 64
8MJ = 5I! 8MJ = 5T! 8MJ = 5U!
40 / 64 30 / 48 20 / 32
8MK = 5U! 8MM = 5T! 8MQ = 5T! 8MS = 5U!
40 / 64 20 / 32 10 / 16 20 / 32
8MT = 5Y! 8MW = 5T! 8MZ = 5U!
80 / 128 20 / 32 20 / 32
8ND = 5Z! 8NE = 5U! 8NF = 5Y! 8NG = 5V!
20 / 32 40 / 64 20 / 32 20 / 32
8NI = 5I! 8NJ = 5I! 8NJ = 5U!
20 / 32 40 / 64 40 / 64
8NJ = 5Z! 8NP = 5Y!
220 / 352 20 / 32
8NQ = 5U! 8NQ = 5W! 8NR = 5L!
20 / 32 20 / 32 40 / 64
8NR = 5Z! 8NS = 5Z! 8NT = 5V!
40 / 64 30 / 48 20 / 32
8NU = 5T! 8NU = 5Z! 8NW = 5V!
40 / 64 20 / 32 20 / 32
8NX = 5W! 8NY = 5V! 8OA = 5N! 8OB = 5T!
40 / 64 20 / 32 20 / 32 20 / 32
8OC = 5T!
20 / 32

### Enneominoes and Tetromino I

<Giovanni Resta's Enneominoes and Tetromino I>

9ABB = 4I!9ADJ = 4I!9AIM = 4I9AJP = 4I!
16 / 36 32 / 72 16 / 36 32 / 72
9ATU = 4I!9AWZ = 4I!9AXB = 4I!9AXX = 4I!
32 / 72 16 / 36 16 / 36 16 / 36
9BAY = 4I!9BFG = 4I!9BFS = 4I!9BGZ = 4I
16 / 36 16 / 36 16 / 36 16 / 36
9BJX = 4I!9BKZ = 4I!9BLB = 4I!9BLW = 4I!
16 / 36 16 / 36 16 / 36 16 / 36
9BMS = 4I!9BMW = 4I!9BNV = 4I!
16 / 36 32 / 72 16 / 36

### Enneominoes and Tetromino Q

<Giovanni Resta's Enneominoes and Tetromino Q>

9ABM = 4Q9AJS = 4Q!9AOF = 4Q
24 / 54 32 / 72 24 / 54
9AYP = 4Q9BAR = 4Q!9BXI = 4Q!
16 / 36 32 / 72 16 / 36

### Decominoes and Tetromino L

<Giovanni Resta's Dekaminoes and Tetromino L>

10HKT = 4L
16 / 40

Last revised 2015-07-15.

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Col. George Sicherman [ HOME | MAIL ]