Caged 6 Piece Burr – Notchable

Caged Burr 枠内に組む Caged 6 Piece Burr の解の分布を示す。

Notchable

Notchable のみの構成を示す。枠には、3種類を想定している。すなわち、6×6×6 を単純に穿ったもの、隅の 2×2×2 を削ってすべて厚さ1にしたもの、枠だけのもである。当然、Type-1 で解けるものは Type-2 でも解ける。Type-2 で解けるものは Type-3 でも解ける。
Type-1、Type-2 の枠では、穴が外側に位置してもいいことに注意せよ。

Type-1

XXXXXX | XXXXXX | X....X | X....X | XXXXXX | XXXXXX
XX..XX | XX..XX | X....X | X....X | XX..XX | XX..XX
XX..XX | ...... | ...... | ...... | ...... | XX..XX
XX..XX | ...... | ...... | ...... | ...... | XX..XX
XX..XX | XX..XX | X....X | X....X | XX..XX | XX..XX
XXXXXX | XXXXXX | X....X | X....X | XXXXXX | XXXXXX

Type-2

XXXXXX | XXXXXX | X....X | X....X | XXXXXX | XXXXXX
XX..XX | X....X | X....X | X....X | X....X | XX..XX
XX..XX | ...... | ...... | ...... | ...... | XX..XX
XX..XX | ...... | ...... | ...... | ...... | XX..XX
XX..XX | X....X | X....X | X....X | X....X | XX..XX
XXXXXX | XXXXXX | X....X | X....X | XXXXXX | XXXXXX

Type-3

XXXXXX | X....X | X....X | X....X | X....X | XXXXXX
X....X | ...... | ...... | ...... | ...... | X....X
X....X | ...... | ...... | ...... | ...... | X....X
X....X | ...... | ...... | ...... | ...... | X....X
X....X | ...... | ...... | ...... | ...... | X....X
XXXXXX | X....X | X....X | X....X | X....X | XXXXXX

Type-1、Type-2 で、分解不可能なものを含めた組み合わせは 1,932,278通り(除鏡像968,482通り)、Type-3 では 1,081,675通り(除鏡像542,755通り) ある。穴の数による内訳は以下のとおり。Type-1 には長さ8以上の解はない。

Length 6 (Type-1)
Holes	Combi	Apart	Assm	Unique Solutions					Non-Unique Solutions
Holes	Combi	Apart	Assm	1 move	2	4	5	6	max 1 move	2	3	4	5	6
0	10	18	18	6					4
1	48	98	124	22					26
2	133	563	800	37					96
3	316	2,404	4,687	38					278
4	666	7,959	18,832	40					566	60
5	1,323	24,108	74,824	24					1,009	290
6	2,444	58,166	233,541	3					1,569	872
7	4,141	125,877	626,311						1,930	2,185	26
8	6,085	229,430	1,448,121						2,037	3,875	172	1
9	7,248	361,805	2,672,977		4		2	2	1,678	5,015	543	4
10	7,227	486,763	4,116,540				4		986	5,170	1,027	28	4	8
11	6,108	561,976	5,067,959				6		513	4,053	1,416	88	12	20
12	4,174	551,617	4,826,210			1	4		235	2,235	1,436	193	23	47
13	2,345	459,445	3,555,657				6		74	857	1,044	266	28	70
14	1,111	324,198	2,022,785						11	237	513	242	33	75
15	464	191,768	915,702						10	36	180	146	26	66
16	172	91,558	316,804						2	5	48	59	13	45
17	56	36,040	83,218								8	24	6	18
18	16	9,886	15,270								3	6		7
19	4	1,592	1,864									2		2
20	1	89	93									1
Total	44,092	3,525,360	26,002,337	170	4	1	22	2	11,024	24,890	6,416	1,060	145	358
MaxMoves				1.4.4.4	2.3.4.3	4.6.3.4	5.5.1.2 .2.3	6.5.1.2 .2.3	1.11	2.13.3.1 .2	3.13.3.1 .2	4.13.2.1 .2	5.5.1.2 .2.3	6.16.1.1 .2
MaxApart									320	649	1,773	1,834	528	2,297
MaxAssm				17	232	89	930	72	8,441	11,024	11,488	11,708	3,320	5,753

Length 6 (Type-2)
Holes	Combi	Apart	Assm	Unique Solutions			Non-Unique Solutions
Holes	Combi	Apart	Assm	1 move	4	6	max 1 move	2	3	4	5	6	7	8	9	10	11	12	13
0	10	18	18	6			4
1	48	98	124	22			26
2	133	563	800	37			96
3	316	2,410	4,687	38			278
4	670	7,985	18,864	44			566	60
5	1,335	24,278	75,032	28			1,017	290
6	2,460	58,780	234,731	5			1,579	876
7	4,173	127,845	632,745				1,956	2,191	26
8	6,179	234,707	1,467,279				2,083	3,917	178	1
9	7,454	373,029	2,738,305		2	2	1,792	5,077	571	8	2
10	7,525	507,655	4,252,462		4		1,138	5,215	1,104	48	8	8
11	6,432	593,577	5,303,465		8	4	637	4,079	1,536	130	18	20
12	4,478	589,097	5,129,838		15	4	337	2,230	1,555	253	27	57
13	2,517	497,883	3,779,575		6		106	823	1,124	334	34	86		4
14	1,159	353,681	2,107,629				12	201	521	269	33	77	8	14	6	8	8	2
15	478	209,248	936,098				6	34	158	142	20	50	4	18	10	12	20	4
16	172	100,460	316,804					3	42	40	6	21	2	13	8	12	18	6	1
17	56	38,990	83,218						4	10		2		8	2	10	8	2	10
18	16	10,412	15,270						2	5				1		1	3		4
19	4	1,654	1,864													2	2
20	1	89	93							1
Total	45,616	3,732,459	27,098,901	180	35	10	11,633	24,996	6,821	1,241	148	321	14	58	26	45	59	14	15
MaxMoves				1.11.2.3	4.12.2 .1.2.2	6.12.2 .1.2.2	1.14.1 .2.2.2	2.13.3	3.13.3	4.12.2 .3	5.5.3 .1.2.3	6.16	7.12.2 .3	8.9.2 .2.2	9.10.4 .2.3.2	10.9.1 .2.2	11.7.4	12.7.4 .4	13.8.1 .3
MaxApart							368	656	1,418	1,306	398	1,264	884	932	1,509	2,365	1,475	1,360	1,882
MaxAssm				28	914	507	6,260	11,024	11,488	8,150	2,891	4,446	4,201	3,978	8,441	11,708	8,565	5,344	4,724

Length 8 (Type-2)
Holes	Combi	Apart	Assm	Unique Solutions	Non-Unique Solutions
Holes	Combi	Apart	Assm	1 move	max 1 move	2	3	4
0	10	18	18	6	4
1	48	98	124	22	26
2	133	563	800	37	96
3	316	2,398	4,687	38	278
4	662	7,941	18,806	38	564	60
5	1,317	23,946	74,790	18	1,009	290
6	2,436	57,728	232,937	3	1,561	872
7	4,125	124,095	624,845		1,914	2,185	26
8	6,032	225,260	1,442,239		1,995	3,865	172
9	7,134	351,687	2,653,577		1,590	5,005	539
10	7,079	467,900	4,076,524		914	5,141	1,020	4
11	5,900	531,197	4,969,173		431	4,049	1,408	12
12	3,967	509,942	4,672,821		187	2,317	1,437	26
13	2,215	414,643	3,418,497		72	1,011	1,102	30
14	1,070	283,142	1,966,783		30	374	638	28
15	452	162,806	902,790		16	116	300	20
16	172	75,725	316,804		11	33	122	6
17	56	28,672	83,218		4	8	42	2
18	16	7,884	15,270			5	11
19	4	1,388	1,864			2	2
20	1	87	93			1
Total	43,145	3,277,120	25,476,660	162	10,702	25,334	6,819	128
MaxMoves				1.1.5.1.2.2	1.4.4.4.2	2.3.1.3.2.2	3.2.1.3.2.2	4.1.1.3.2.2
MaxApart					252	851	1,920	1,090
MaxAssm				17	8,441	11,024	11,708	3,908

Length 6 (Type-3)
Holes	Combi	Apart	Assm	Unique Solutions														Non-Unique Solutions
Holes	Combi	Apart	Assm	1 move	2	3	4	5	6	7	8	9	10	11	12	13	22	max 1 move	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26
0	22	37	40	11														11
1	132	262	318	62														70
2	485	1,314	1,990	210														261	14
3	1,100	5,474	9,111	256														746	92	6
4	1,998	17,274	33,704	116	20	3												1,413	429	17
5	4,167	47,410	114,188	38	84	12												2,345	1,566	116			6
6	9,027	107,677	314,083	100	310	59		10	4									3,894	4,091	501	2	8	46	2
7	17,017	208,810	766,935	110	602	172	2	40	18									5,561	8,798	1,500	16	64	132		2
8	23,723	343,494	1,467,923	37	918	195		132	34									5,680	13,194	2,882	86	304	235	21	5
9	27,384	478,290	2,349,600	24	1,022	210		256	42	34		12	10		2			4,372	15,390	4,136	308	1,004	362	96	60	28	14	2
10	24,495	576,490	2,895,119	8	551	96	4	260	18	34	16	24	22	16	4		2	2,139	12,393	4,659	533	1,832	720	525	230	222	118	18	33	6	6		2	2	2
11	15,748	617,520	2,541,108	2	122		6	66	4	2	6	10	8	4	2	2		499	5,400	3,153	490	1,960	968	924	508	550	468	168	208	54	66	26	22	10	26	4	4		4				2
12	7,951	583,983	1,687,424		20			12										80	1,437	1,026	182	972	554	744	492	558	529	304	387	141	150	113	91	52	67	8	12	8	6			2	4
13	3,477	476,254	909,306		4													19	394	214	62	244	256	290	290	350	282	164	236	110	98	144	106	66	74	24	18	8	10	4		4	6
14	1,391	307,499	405,557															8	164	74	18	75	140	83	151	104	66	73	89	58	45	86	69	26	20	16	10	4		4	2		6
15	510	138,938	147,984															4	108	50	6	30	56	32	36	44	14	34	14	28	18	8	8	2	6	2				4	2		4
16	177	42,299	42,533															1	82	40			17	9	10	4	2	4	4	4
17	56	9,256	9,256																42	10			4
18	16	1,368	1,368																16
19	4	128	128																4
20	1	5	5																1
Total	138,881	3,963,782	13,697,680	974	3,653	747	12	776	120	70	22	46	40	20	8	2	2	27,103	63,615	18,384	1,703	6,493	3,496	2,726	1,784	1,860	1,493	767	971	401	383	377	298	158	195	54	44	20	20	12	4	6	22
MaxMoves				1.18.2 .4.2.2	2.8.2	3.9.2 .5.3.2	4.10.4 .2.2.2	5.11.4 .6.2.2	6.11.2 .1.2.2	7.9.4 .4.2.2	8.5.1 .3.2	9.3.2 .3	10.4.2 .3	11.3.2 .3	12.2.2 .6.2.2	13.2.2 .2.2	22.1.2 .2	1.18.2 .4.2.2	2.17.3 .4.2.2	3.13.2 .2.2.2	4.12.3 .4.2.2	5.17.3 .2.2.2	6.14.1 .2.2	7.15.3 .3.3.2	8.15.1 .2.2	9.17.3 .5.2	10.16.3 .6.2	11.14.3 .4	12.14.3 .3.2	13.14.3 .3.2	14.16.3 .3.2.2	15.9	16.8.2 .2.2.2	17.6.3 .2.2.2	18.6.3 .3.2.2	19.6.2 .2.2	20.5.2 .2.2	21.3.3 .4.2.2	22.7.1 .2	23.4.3 .4.2	24.6.2 .5.3.2	25.2.2	26.4.2 .2
MaxApart																		896	452	816	496	359	1,916	768	960	967	785	782	745	1,482	1,963	1,071	1,781	1,068	1,860	1,062	992	571	417	616	920	79	882
MaxAssm				72	356	134	56	1,536	768	268	224	152	192	72	114	30	1	992	1,536	920	1,824	1,824	2,448	1,920	1,344	1,376	2,464	1,536	1,536	2,448	3,072	2,448	2,464	1,920	3,072	1,536	1,728	880	1,152	960	960	192	960

Length 8 (Type-3)
Holes	Combi	Apart	Assm	Unique Solutions							Non-Unique Solutions
Holes	Combi	Apart	Assm	1 move	2	3	4	5	7	11	max 1 move	2	3	4	5	6	7	8	9	10	11	12	13
0	21	36	39	10							11
1	98	210	238	42							56
2	318	992	1,457	124							181	13
3	743	3,966	6,753	168	2						503	64	6
4	1,476	12,145	24,667	132	24						962	341	17
5	3,002	32,248	83,322	48	88						1,566	1,196	104
6	6,154	72,525	232,364	36	266	26					2,261	3,171	394
7	11,314	141,324	565,847	20	492	56					2,814	6,738	1,190	4
8	16,288	235,435	1,115,022	21	628	54		4			2,758	10,473	2,308	36	4	2
9	19,108	336,306	1,805,556	10	656	82		8			2,060	12,940	3,176	144	20	12
10	17,995	412,829	2,319,015	8	317	46	4	12	2		1,103	12,591	3,493	280	78	57	4
11	12,874	440,406	2,186,822	4	96	4	6	42	6		393	8,596	2,943	442	140	178	20	2	2
12	7,246	405,659	1,556,485		6			6		2	104	4,197	1,775	463	278	305	78	12	10	2	4		4
13	3,361	315,186	880,450					2			19	1,572	712	334	238	306	102	34	16	4	14	2	6
14	1,379	201,159	403,509								8	526	231	136	145	170	80	51	16	2	10	4
15	510	102,534	147,984								4	186	96	36	40	66	32	42	8
16	177	37,929	42,533								1	92	52	6	4		10	12
17	56	9,190	9,256									42	10				4
18	16	1,368	1,368									16
19	4	128	128									4
20	1	5	5									1
Total	102,141	2,761,580	11,382,820	623	2,575	268	10	74	8	2	14,804	62,759	16,507	1,881	947	1,096	330	153	52	8	28	6	10
MaxMoves				1.9	2.12.2 .2	3.5.3	4.1.3 .6	5.8.2 .2.2	7.4	11.5.2	1.11.2 .6.2.2	2.12.4 .6.2.2	3.11.2 .2	4.5.1 .2.2	5.11.4 .8	6.12.4 .5.2.2	7.5.2	8.8.2	9.9.2	10.8.2	11.5.2	12.4.1 .3.2	13.4.1 .3.2
MaxApart											848	448	1,804	1,120	1,116	1,068	396	1,004	667	338	130	178	263
MaxAssm				96	260	134	26	192	192	68	1,056	3,072	3,072	1,536	1,536	1,920	960	1,920	768	360	384	184	280

Length 10 (Type-3)
Holes	Combi	Apart	Assm	Unique Solutions				Non-Unique Solutions
Holes	Combi	Apart	Assm	1 move	2	3	4	max 1 move	2	3	4	5
0	18	30	30	10				8
1	86	182	204	40				46
2	269	829	1,147	105				152	12
3	632	3,368	5,583	132				444	52	4
4	1,255	10,207	20,036	98	12			847	285	13
5	2,543	27,012	67,772	42	46			1,401	978	76
6	5,116	60,439	188,625	32	160	20		2,065	2,522	317
7	9,428	117,408	466,641	22	332	52		2,570	5,398	1,050	4
8	13,633	195,747	927,319	11	450	44		2,578	8,471	2,045	32	2
9	16,164	279,528	1,537,388	2	542	72		2,044	10,548	2,814	136	6
10	15,608	344,611	2,045,852	2	310	46	2	1,278	10,551	3,125	268	26
11	11,364	369,396	1,984,586		92	8	4	569	7,576	2,647	426	42
12	6,565	341,890	1,456,254		14			174	4,179	1,647	479	72
13	3,157	268,066	843,002					41	1,952	716	378	70
14	1,340	171,933	393,493					13	820	277	188	42
15	504	88,162	146,960					4	292	130	62	16
16	177	33,773	42,533					1	112	54	6	4
17	56	8,736	9,256						46	10
18	16	1,368	1,368						16
19	4	128	128						4
20	1	5	5						1
Total	87,936	2,322,818	10,138,182	496	1,958	242	6	14,235	53,815	14,925	1,979	280
MaxMoves				1.8.4	2.6	3.5.3	4.1.2	1.8.4.2	2.8	3.7.1.2.2	4.5.1.2.2	5.6.3
MaxApart								784	828	1,688	1,054	1,050
MaxAssm				83	456	134	26	1,224	3,072	3,072	1,536	1,536

Sep 20, 2002 by k16@chiba.email.ne.jp