All Paterson’s Worms

The table below has details on all 412 of Paterson's worms. Of the 412 worms, 336 are known to finish, 73 are known to be infinite, 2 are almost certain (but not proven) to be infinite, and one remains undecided.

Explanation of columns:

• Pattern number: this is simply a name, assigned in the order the worms are encountered in a simple depth-first search.

• Rule: Each digit in the rule corresponds to a direction, relative to the worm's current orientation; 0 is straight forward, and the other 5 directions are numbered counter-clockwise, with 1 being a 60° left turn, 2 being a 120° left turn, etc. Direction 3 (straight backward) never appears since the worm never retraces its path, and of course the path behind it is always taken (it's where the worm just arrived from). The worm starts with a single step forward, and then looks around and sees five untaken paths around it. It chooses one of these directions, and this becomes the first digit of the rule; every time in the future that it sees five untaken paths, it will choose the same path. The first time it sees some other configuration of taken and untaken paths, it makes a new decision and this becomes the second digit of its rule. In other words, the rules below specify what decision the worm should make when it encounters a new situation, and these are given in the order that the decisions will be made.

• Pictures: Small worms have just a single picture, showing the individual lines. The green circle marks the origin, and the first step of the worm is always to the right. Some number of the most recent steps are drawn in red. Most infinite worms, and some interesting finite ones, have a second zoomed-out picture which shows the larger structure. The color changes gradually over time to give some indication of the order in which steps were taken (and to make it look nice). These pictures show the number of steps the worm has taken, and the scale; if the scale is 4, then each pixel represents a 4x4 block of the underlying grid. Some of the bigger worms have links to separate albums with many pictures.

More details on the interesting cases are on the main Paterson's worms page.

Pattern number	Rule	Length / Lower Bound	Pictures (click for larger version)		Pattern number	Rule	Length / Lower Bound	Pictures (click for larger version)
001	0	Infinite			207	1421225	46
002	1010001	295			208	1421251	29
003	1010002	105			209	1421254	52
004	1010005	67			210	1421255	Infinite
005	101001	33			211	1421411	Infinite
006	1010051	2,155			212	1421412	29
007	1010052	235			213	1421415	35
008	1010055	1,932			214	142144	Infinite
009	101020	86			215	142145	Infinite
010	101021	99			216	14251	48
011	1010251	327			217	14252	Infinite
012	1010252	475			218	14255	44
013	1010255	61			219	1450011	83
014	1010401	68			220	1450012	81
015	1010402	196			221	1450015	94
016	1010405	67			222	1450041	Infinite
017	101041	48			223	1450042	897
018	1010451	1,831			224	1450045	285
019	1010452	684			225	1450051	90
020	1010455	99			226	1450052	73
021	1015001	85			227	1450055	697
022	1015002	83			228	1450211	Infinite
023	1015005	96			229	1450214	Infinite
024	101501	48			230	1450215	Infinite
025	1015051	7,524			231	1450221	Infinite
026	1015052	203			232	1450224	16,811,365,528	Paterson's worm 232
027	1015055	67			233	1450225	284
028	1015201	10,460			234	145025	34
029	1015202	Infinite			235	145041	90
030	1015205	Infinite			236	1450441	50
031	101521	29			237	1450442	Infinite
032	1015251	534			238	1450445	1,688
033	1015252	4,318			239	1450451	4,419
034	1015255	93			240	1450452	3,566
035	1015410	66			241	1450455	496
036	1015411	Infinite			242	145101	29
037	1015415	534			243	1451041	50
038	1015420	43			244	1451042	248
039	1015421	48			245	1451045	39
040	1015425	4,432			246	1451051	73
041	1015450	53			247	1451052	5,715
042	1015451	48			248	1451055	Infinite
043	1015455	67			249	1451211	33
044	1040010	54			250	1451214	219
045	1040012	296			251	1451215	79
046	1040020	1,660			252	1451221	23
047	1040022	970			253	1451224	3,793
048	1040050	71			254	1451225	46
049	1040052	102			255	1451251	29
050	1040101	289			256	1451254	52
051	1040102	97			257	1451255	Infinite
052	1040105	110			258	1451411	Infinite
053	104015	57			259	1451412	29
054	1040510	1,545			260	1451415	35
055	1040512	569,804			261	145144	Infinite
056	1040521	1020			262	145145	Infinite
057	1040524	10,795			263	14551	48
058	1040550	451			264	14552	Infinite
059	1040555	451			265	14555	44
060	1042010	918,339			266	150011	50
061	1042015	Unknown - lower bound 1.4 * 10^21	Paterson's worm 061		267	150015	33
062	1042020	57,493,855,205,939	Paterson's worm 062		268	1500411	196
063	1042022	57,493,855,205,905			269	1500415	69
064	104205	Infinite			270	1500421	732
065	1042110	631			271	1500425	159
066	1042115	118			272	1500451	78
067	1042120	2,056			273	1500455	45
068	1042122	747			274	15005	Infinite
069	1042150	5,865			275	150111	33
070	1042155	398			276	150115	29
071	1042501	1,550			277	1501411	55
072	1042502	130			278	1501412	71
073	1042505	153			279	1501415	44
074	1042521	2,754			280	1501451	140
075	1042522	196			281	1501452	113
076	1042525	122			282	1501455	119
077	1044001	Infinite			283	1501511	1,672
078	1044002	Infinite			284	1501515	10,307
079	1044005	46			285	1501521	Infinite
080	1044011	Infinite			286	1501525	165
081	1044012	Infinite			287	1501551	134
082	1044015	212			288	1501555	112
083	1044051	330			289	1505111	52,549
084	1044052	Infinite			290	1505115	Infinite
085	1044055	138			291	1505141	1,632
086	1044201	Infinite			292	1505145	1,632
087	1044202	Infinite			293	1505151	1,978
088	1044205	17,859			294	1505155	5,148
089	104421	Infinite			295	1505211	419
090	1044251	347			296	1505215	1,564
091	1044252	Infinite			297	1505241	4,371
092	1044255	73			298	1505245	7,143
093	1051	28			299	1505251	454
094	1054	28			300	1505255	306
095	120012	110			301	1505511	62
096	1200141	247			302	1505514	130
097	1200142	309			303	1505515	55
098	1200145	181			304	1505551	636
099	1200412	Infinite			305	1505554	206
100	1200414	711			306	1505555	242
101	1200422	5,132			307	1520	Infinite
102	1200424	345			308	152111	33
103	1200452	1,515			309	152115	29
104	1200454	839			310	1521411	148
105	1200521	304			311	1521415	247
106	1200522	354			312	1521421	393
107	1200525	631			313	1521425	113
108	1200541	3,943			314	1521451	136
109	1200542	201			315	1521455	132
110	1200545	237			316	1521511	623
111	1202	Infinite			317	1521515	2,377
112	1204141	176			318	1521521	176
113	1204142	735			319	1521525	1,742
114	1204145	92			320	1521551	152
115	120415	57			321	1521555	112
116	1204412	415			322	1525111	534
117	1204414	Infinite			323	1525115	Infinite	Paterson's worm 323
118	120442	Infinite			324	1525121	356
119	1204454	2578			325	1525125	606
120	1204455	2578			326	1525151	533
121	12045	Infinite			327	1525155	287
122	121401	48			328	1525411	2,754
123	1214041	77			329	1525412	Infinite
124	1214042	609			330	1525415	126
125	1214045	114			331	1525451	5,708
126	1214051	22,847			332	1525452	Infinite
127	1214052	Infinite			333	1525455	338
128	1214055	134			334	1525511	83,618
129	121421	29			335	1525515	1574
130	1214241	105			336	1525521	7,275
131	1214242	7,882			337	1525525	843
132	1214245	45			338	1525551	262
133	1214251	281			339	1525555	286
134	1214252	62			340	1541011	194
135	1214255	152			341	1541012	294
136	1214411	Infinite			342	1541015	180
137	1214412	48			343	1541051	354
138	1214415	48			344	1541052	107
139	121444	Infinite			345	1541055	708
140	121445	Infinite			346	154111	Infinite
141	121510	33			347	1541121	48
142	121512	99			348	1541125	114
143	121514	48			349	1541151	48
144	1215401	151			350	1541155	58
145	1215402	42			351	1541511	2,811
146	1215405	53			352	1541515	663
147	1215421	2,526			353	1541521	Infinite
148	1215422	2,857			354	1541525	Infinite
149	1215425	561			355	1541551	245
150	121544	33			356	1541555	321
151	12155	28			357	1544101	Infinite
152	1251	48			358	1544102	Infinite
153	1252101	220,142			359	1544105	45,477
154	1252104	7,584			360	154411	Infinite
155	1252105	9,260			361	1544151	558
156	1252121	Infinite	Paterson's worm 156		362	1544152	Infinite
157	1252124	2,565			363	1544155	7,565
158	1252125	731			364	1544501	Infinite
159	1252141	553			365	1544502	Infinite
160	1252144	173			366	1544505	512
161	1252145	Infinite			367	1544511	Infinite
162	12522	Infinite			368	1544512	Infinite
163	1252501	248			369	1544515	1,457
164	1252504	236			370	1544551	4,802
165	1252505	101			371	1544552	Infinite
166	1252521	1,063			372	1544555	138
167	1252524	162			373	1545	Infinite
168	1252525	86			374	200	9
169	1252541	514			375	2010	15
170	1252544	1,116			376	20140	30
171	1252545	738			377	201412	45
172	1420011	83			378	2014141	609
173	1420012	81			379	2014142	3,563,608,205	Paterson's worm 379
174	1420015	94			380	2014144	615
175	1420041	Infinite			381	201415	37
176	1420042	897			382	20142	27
177	1420045	285			383	2015	Infinite
178	1420051	90			384	2104	12
179	1420052	73			385	2105	21
180	1420055	697			386	2144	18
181	1420211	Infinite			387	21450	63
182	1420214	Infinite			388	2145121	438
183	1420215	Infinite			389	2145122	438
184	1420221	Infinite			390	2145124	411
185	1420224	16,811,365,528	Paterson's worm 185		391	2145141	2,478
186	1420225	284			392	2145142	87,996,218	Paterson's worm 392
187	142025	34			393	2145144	2,373
188	142041	90			394	214515	68
189	1420441	50			395	21452	48
190	1420442	Infinite			396	215	Infinite
191	1420445	1,688			397	244	9
192	1420451	4,419			398	2450	15
193	1420452	3,566			399	24540	30
194	1420455	496			400	245412	45
195	142101	29			401	2454141	609
196	1421041	50			402	2454142	3,563,608,205	Paterson's worm 402
197	1421042	248			403	2454144	615
198	1421045	39			404	245415	37
199	1421051	73			405	24542	27
200	1421052	5,715			406	2455	Infinite
201	1421055	Infinite			407	2510	12
202	1421211	33			408	2514	18
203	1421214	219			409	2515	Infinite
204	1421215	79			410	2540	12
205	1421221	23			411	2544	18
206	1421224	3,793			412	2545	Infinite