Author:	David Goodger <goodger@python.org>
Date:	2015-02-24
Revision:	600
Web site:	http://puzzler.sourceforge.net/
Copyright:	© 1998-2015 by David J. Goodger
License:	GPL 2

Polyform Counts

Units are squares.

Order	Name	Free	Units	Sum	One-sided	Units	Sum
1	monomino	1	1	1	1	1	1
2	domino	1	2	3	1	2	3
3	tromino	2	6	9	2	6	9
4	tetromino	5	20	29	7	28	37
5	pentomino	12	60	89	18	90	127
6	hexomino	35	210	299	60	360	487
7*	heptomino	108	756*	1055	196	1014*	1501
8*	octomino	369	2952*	4007	704	4928*	6429
9*	enneomino	1285	11565*		2500	22500*
10*	decomino	4655	46550*		9189	91890*

"*" indicates that pieces with holes exist.

Shapes

Squares:

S(n) = n²

Triangles (n == height):

T(n) = n(n + 1) / 2

#
##
###
####

Diamonds (n == side length or height of quadrant; A001844):

D(n) = 4T(n) - 4n +1
     = 2n² - 2n + 1
     = 2n(n - 1) + 1
     = n² + (n - 1)²
    (== centered square numbers)

  #
 ###
#####
 ###
  #

Aztec Diamonds (n == side length or height of quadrant; A046092):

A(n) = 2n(n + 1)

  ##
 ####
######
######
 ####
  ##

Double-Triangles (n == height; base = 2n-1):

DT(n) = n²
      = S(n)

  #
 ###
#####

n	S	T	D	A
1	1	1	1	4
2	4	3	5	12
3	9	6	13	24
4	16	10	25	40
5	25	15	41	60
6	36	21	61	84
7	49	28	85	112
8	64	36	113	144
9	81	45	145	180
10	100	55	181	220
11	121	66	221	264
12	144	78	265	312
13	169	91	313	364
14	196	105	365	420
15	225	120	421	480
16	256	136	481	544
17	289	153	545	612
18	324	171	613	684
19	361	190	685	760
20	400	210	761	840
21	441	231	841	924
22	484	253	925	1012
23	529	276	1013	1104
24	576	300	1105	1200
25	625	325	1201	1300
26	676	351	1301	1404
27	729	378	1405	1512
28	784	406	1513	1624
29	841	435	1625	1740
30	900	465	1741	1860
31	961	496	1861	1984
32	1024	528	1985	2112
33	1089	561	2113	2244
34	1156	595	2245	2380
35	1225	630	2381	2520
36	1296	666	2521	2664
37	1369	703	2665	2812
38	1444	741	2813	2964
39	1521	780	2965	3120
40	1600	820	3121	3280

Rectangles:

R(m,n) = mn

R	m=1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
n=1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
2	...	4	6	8	10	12	14	16	18	20	22	24	26	28	30
3	...	...	9	12	15	18	21	24	27	30	33	36	39	42	45
4	...	...	...	16	20	24	28	32	36	40	44	48	52	56	60
5	...	...	...	...	25	30	35	40	45	50	55	60	65	70	75
6	...	...	...	...	...	36	42	48	54	60	66	72	78	84	90
7	...	...	...	...	...	...	49	56	63	70	77	84	91	98	105
8	...	...	...	...	...	...	...	64	72	80	88	96	104	112	120
9	...	...	...	...	...	...	...	...	81	90	99	108	117	126	135
10	...	...	...	...	...	...	...	...	...	100	110	120	130	140	150
11	...	...	...	...	...	...	...	...	...	...	121	132	143	154	165
12	...	...	...	...	...	...	...	...	...	...	...	144	156	168	180
13	...	...	...	...	...	...	...	...	...	...	...	...	169	182	195
14	...	...	...	...	...	...	...	...	...	...	...	...	...	196	210
15	...	...	...	...	...	...	...	...	...	...	...	...	...	...	225

R	m=16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
n=1	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
2	32	34	36	38	40	42	44	46	48	50	52	54	56	58	60
3	48	51	54	57	60	63	66	69	72	75	78	81	84	87	90
4	64	68	72	76	80	84	88	92	96	100	104	108	112	116	120
5	80	85	90	95	100	105	110	115	120	125	130	135	140	145	150
6	96	102	108	114	120	126	132	138	144	150	156	162	168	174	180
7	112	119	126	133	140	147	154	161	168	175	182	189	196	203	210
8	128	136	144	152	160	168	176	184	192	200	208	216	224	232	240
9	144	153	162	171	180	189	198	207	216	225	234	243	252	261	270
10	160	170	180	190	200	210	220	230	240	250	260	270	280	290	300
11	176	187	198	209	220	231	242	253	264	275	286	297	308	319	330
12	192	204	216	228	240	252	264	276	288	300	312	324	336	348	360
13	208	221	234	247	260	273	286	299	312	325	338	351	364	377	390
14	224	238	252	266	280	294	308	322	336	350	364	378	392	406	420
15	240	255	270	285	300	315	330	345	360	375	390	405	420	435	450

R	m=16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
n=16	256	272	288	304	320	336	352	368	384	400	416	432	448	464	480
17	...	289	306	323	340	357	374	391	408	425	442	459	476	493	510
18	...	...	324	342	360	378	396	414	432	450	468	486	504	522	540
19	...	...	...	361	380	399	418	437	456	475	494	513	532	551	570
20	...	...	...	...	400	420	440	460	480	500	520	540	560	580	600
21	...	...	...	...	...	441	462	483	504	525	546	567	588	609	630
22	...	...	...	...	...	...	484	506	528	550	572	594	616	638	660
23	...	...	...	...	...	...	...	529	552	575	598	621	644	667	690
24	...	...	...	...	...	...	...	...	576	600	624	648	672	696	720
25	...	...	...	...	...	...	...	...	...	625	650	675	700	725	750
26	...	...	...	...	...	...	...	...	...	...	676	702	728	754	780
27	...	...	...	...	...	...	...	...	...	...	...	729	756	783	810
28	...	...	...	...	...	...	...	...	...	...	...	...	784	812	840
29	...	...	...	...	...	...	...	...	...	...	...	...	...	841	870
30	...	...	...	...	...	...	...	...	...	...	...	...	...	...	900

Potential Puzzles

Puzzles not otherwise noted below have not been implemented or solved.

Initial numbers are the counts of unit squares in the puzzles.

20 unit squares: tetrominoes -- no symmetrical puzzles due to parity (T4)

28: one-sided tetrominoes -- no symmetrical puzzles due to parity (T4)

28: polyominoes of order 2 - 4

R(7,4)

29: polyominoes of order 1 - 4

Potential:
No solutions:
- R(5,5) + R(9,1) + R(1,9) - 4 squares (= Polyominoes1234Cross_x)

36: one-sided polyominoes of order 2 - 4

37: one-sided polyominoes of order 1 - 4

60 unit squares: pentominoes

Potential:
- heart shape?
- ring/tube versions of 3x20, 4x15, 5x12, 6x10 (joined at short or long edges)
- toroidal versions of 3x20, 4x15, 5x12, 6x10
- Möbius strip versions of 3x20, 4x15, 5x12, 6x10 (joined at short or long edges)
- Klein bottle?
No solutions:
- 2R(10,4) crossed - S(2) (= PentominoesCross_X1)
- 2R(12,2) crossed + 2R(8,4) crossed overlayed - S(2) (= PentominoesCross_X2)
- 2R(10,4) crossed - 4 units (= PentominoesCross_X3)
- 2R(9,5) crossed - 5 units in "X" (= PentominoesCross_X4)
- T(14) - T(9) (= PentominoesTrapezoid_X1)
- staircase chevron (= PentominoesChevron_X)
- 10x6 skew (10 columns of 6 rows each, offset by one each, like a staircase; = PentominoesSkewed_x1)
- D(7) - D(4) (= PentominoesDiamondV_x)
- A(5)

61: pentominoes + monomino

64: pentominoes + 1 tetromino (square)

No solutions:
- D(6) + R(1,3) (as a "lollipop stick"; = PentominoesPlusSquareTetrominoDiamond_x1)

80: polyominoes of order 4 & 5 -- tetrominoes + pentominoes

88: polyominoes of order 2 - 5

R(11,8)

89: polyominoes of order 1 - 5

90: one-sided pentominoes

126: one-sided polyominoes of order 2 - 5

127: one-sided polyominoes of order 1 - 5

210: hexominoes -- no simple symmetrical puzzles due to parity

216: hexominoes + 1 duplicate == 6³ == Kadon "Sextillions"

298: polyominoes of order 2 - 6

299: polyominoes of order 1 - 6

"polystar" (http://gamepuzzles.com/polystar.htm, solution by Darian Jenkins, suggested by Dan Klarskov, email 2012-03-14)

360: one-sided hexominoes

486: one-sided polyominoes of order 2 - 6

487: one-sided polyominoes of order 1 - 6

Notes on Polyominoes

Polyform Counts

Shapes

Potential Puzzles

Links

R	m=1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
n=1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
2	...	4	6	8	10	12	14	16	18	20	22	24	26	28	30
3	...	...	9	12	15	18	21	24	27	30	33	36	39	42	45
4	...	...	...	16	20	24	28	32	36	40	44	48	52	56	60
5	...	...	...	...	25	30	35	40	45	50	55	60	65	70	75
6	...	...	...	...	...	36	42	48	54	60	66	72	78	84	90
7	...	...	...	...	...	...	49	56	63	70	77	84	91	98	105
8	...	...	...	...	...	...	...	64	72	80	88	96	104	112	120
9	...	...	...	...	...	...	...	...	81	90	99	108	117	126	135
10	...	...	...	...	...	...	...	...	...	100	110	120	130	140	150
11	...	...	...	...	...	...	...	...	...	...	121	132	143	154	165
12	...	...	...	...	...	...	...	...	...	...	...	144	156	168	180
13	...	...	...	...	...	...	...	...	...	...	...	...	169	182	195
14	...	...	...	...	...	...	...	...	...	...	...	...	...	196	210
15	...	...	...	...	...	...	...	...	...	...	...	...	...	...	225

R	m=16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
n=1	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
2	32	34	36	38	40	42	44	46	48	50	52	54	56	58	60
3	48	51	54	57	60	63	66	69	72	75	78	81	84	87	90
4	64	68	72	76	80	84	88	92	96	100	104	108	112	116	120
5	80	85	90	95	100	105	110	115	120	125	130	135	140	145	150
6	96	102	108	114	120	126	132	138	144	150	156	162	168	174	180
7	112	119	126	133	140	147	154	161	168	175	182	189	196	203	210
8	128	136	144	152	160	168	176	184	192	200	208	216	224	232	240
9	144	153	162	171	180	189	198	207	216	225	234	243	252	261	270
10	160	170	180	190	200	210	220	230	240	250	260	270	280	290	300
11	176	187	198	209	220	231	242	253	264	275	286	297	308	319	330
12	192	204	216	228	240	252	264	276	288	300	312	324	336	348	360
13	208	221	234	247	260	273	286	299	312	325	338	351	364	377	390
14	224	238	252	266	280	294	308	322	336	350	364	378	392	406	420
15	240	255	270	285	300	315	330	345	360	375	390	405	420	435	450

R	m=1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
n=1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
2	...	4	6	8	10	12	14	16	18	20	22	24	26	28	30
3	...	...	9	12	15	18	21	24	27	30	33	36	39	42	45
4	...	...	...	16	20	24	28	32	36	40	44	48	52	56	60
5	...	...	...	...	25	30	35	40	45	50	55	60	65	70	75
6	...	...	...	...	...	36	42	48	54	60	66	72	78	84	90
7	...	...	...	...	...	...	49	56	63	70	77	84	91	98	105
8	...	...	...	...	...	...	...	64	72	80	88	96	104	112	120
9	...	...	...	...	...	...	...	...	81	90	99	108	117	126	135
10	...	...	...	...	...	...	...	...	...	100	110	120	130	140	150
11	...	...	...	...	...	...	...	...	...	...	121	132	143	154	165
12	...	...	...	...	...	...	...	...	...	...	...	144	156	168	180
13	...	...	...	...	...	...	...	...	...	...	...	...	169	182	195
14	...	...	...	...	...	...	...	...	...	...	...	...	...	196	210
15	...	...	...	...	...	...	...	...	...	...	...	...	...	...	225

R	m=16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
n=1	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
2	32	34	36	38	40	42	44	46	48	50	52	54	56	58	60
3	48	51	54	57	60	63	66	69	72	75	78	81	84	87	90
4	64	68	72	76	80	84	88	92	96	100	104	108	112	116	120
5	80	85	90	95	100	105	110	115	120	125	130	135	140	145	150
6	96	102	108	114	120	126	132	138	144	150	156	162	168	174	180
7	112	119	126	133	140	147	154	161	168	175	182	189	196	203	210
8	128	136	144	152	160	168	176	184	192	200	208	216	224	232	240
9	144	153	162	171	180	189	198	207	216	225	234	243	252	261	270
10	160	170	180	190	200	210	220	230	240	250	260	270	280	290	300
11	176	187	198	209	220	231	242	253	264	275	286	297	308	319	330
12	192	204	216	228	240	252	264	276	288	300	312	324	336	348	360
13	208	221	234	247	260	273	286	299	312	325	338	351	364	377	390
14	224	238	252	266	280	294	308	322	336	350	364	378	392	406	420
15	240	255	270	285	300	315	330	345	360	375	390	405	420	435	450

R	m=1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
n=1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
2	...	4	6	8	10	12	14	16	18	20	22	24	26	28	30
3	...	...	9	12	15	18	21	24	27	30	33	36	39	42	45
4	...	...	...	16	20	24	28	32	36	40	44	48	52	56	60
5	...	...	...	...	25	30	35	40	45	50	55	60	65	70	75
6	...	...	...	...	...	36	42	48	54	60	66	72	78	84	90
7	...	...	...	...	...	...	49	56	63	70	77	84	91	98	105
8	...	...	...	...	...	...	...	64	72	80	88	96	104	112	120
9	...	...	...	...	...	...	...	...	81	90	99	108	117	126	135
10	...	...	...	...	...	...	...	...	...	100	110	120	130	140	150
11	...	...	...	...	...	...	...	...	...	...	121	132	143	154	165
12	...	...	...	...	...	...	...	...	...	...	...	144	156	168	180
13	...	...	...	...	...	...	...	...	...	...	...	...	169	182	195
14	...	...	...	...	...	...	...	...	...	...	...	...	...	196	210
15	...	...	...	...	...	...	...	...	...	...	...	...	...	...	225

R	m=16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
n=1	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
2	32	34	36	38	40	42	44	46	48	50	52	54	56	58	60
3	48	51	54	57	60	63	66	69	72	75	78	81	84	87	90
4	64	68	72	76	80	84	88	92	96	100	104	108	112	116	120
5	80	85	90	95	100	105	110	115	120	125	130	135	140	145	150
6	96	102	108	114	120	126	132	138	144	150	156	162	168	174	180
7	112	119	126	133	140	147	154	161	168	175	182	189	196	203	210
8	128	136	144	152	160	168	176	184	192	200	208	216	224	232	240
9	144	153	162	171	180	189	198	207	216	225	234	243	252	261	270
10	160	170	180	190	200	210	220	230	240	250	260	270	280	290	300
11	176	187	198	209	220	231	242	253	264	275	286	297	308	319	330
12	192	204	216	228	240	252	264	276	288	300	312	324	336	348	360
13	208	221	234	247	260	273	286	299	312	325	338	351	364	377	390
14	224	238	252	266	280	294	308	322	336	350	364	378	392	406	420
15	240	255	270	285	300	315	330	345	360	375	390	405	420	435	450