Base 67: Iteration Frequencies: Full Cycle

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDEF
GHIJKLMNOPQRSTUV
WXYZabcdefghijkl
mnopqrstuvwxyzΑΒ
ΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣ

Color Key
OrangeFixed Pointnumber that becomes itself after one iteration
GreenFinal Numbersall numbers eventually reached, from all cycles
PurpleCycle Startiterations include entry into cycle, assuming it repeats
RedFull Cycleiterations include entire cycle, proving it repeats
BlueStart Numberssubset (optimization) of all numbers; minimal for full coverage
GrayNo Resultsany calculation that has no results

Digits 1 2 3 4 5 6 7 8
Average Steps
Full Cycle		1.99 step
avg		6.81 step
avg		15.70 step
avg		34.80 step
avg		17.51 step
avg		45.03 step
avg		25.89 step
avg		79.62 step
avg
Step Frequency
Full Cycle
0s:
1s:
2s:
0×
1×
66×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
0×
1×
66×
33×
70×
276×
248×
528×
1,056×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
0×
1×
66×
2,311×
1,152×
2,302×
2,292×
2,278×
2,260×
2,238×
2,212×
2,182×
2,148×
2,110×
2,068×
2,022×
1,972×
1,918×
1,860×
1,798×
1,732×
1,662×
1,588×
1,510×
1,428×
1,342×
1,252×
1,158×
1,060×
958×
852×
742×
628×
510×
388×
262×
132×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
0×
1×
66×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
221,408×
652,784×
41,481×
1,155×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
0×
1×
66×
0×
36,231×
89,078×
78,407×
111,544×
202,893×
262,193×
389,155×
407,629×
602,437×
548,223×
809,984×
720,794×
1,063,255×
891,113×
1,180,200×
890,325×
1,043,664×
768,692×
886,790×
538,988×
541,635×
366,796×
272,801×
169,377×
95,157×
39,673×
10,644×
2,164×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
57s:
58s:
59s:
60s:
61s:
62s:
63s:
64s:
65s:
66s:
67s:
68s:
69s:
70s:
71s:
72s:
73s:
74s:
75s:
76s:
77s:
78s:
79s:
80s:
81s:
82s:
83s:
84s:
85s:
86s:
87s:
88s:
89s:
90s:
91s:
92s:
93s:
94s:
95s:
96s:
97s:
98s:
99s:
100s:
101s:
102s:
103s:
104s:
105s:
106s:
0×
1×
66×
0×
0×
23,360×
61,728×
267,776×
1,142,381×
459,080×
414,921×
589,500×
812,212×
904,796×
1,328,461×
1,172,462×
1,394,895×
1,956,741×
4,835,171×
2,123,741×
1,718,264×
2,055,808×
2,414,241×
2,770,036×
3,048,175×
2,586,499×
2,579,618×
2,027,141×
2,331,979×
2,377,206×
2,675,280×
2,488,525×
2,945,549×
3,062,292×
3,472,792×
3,582,275×
4,083,336×
4,008,816×
4,067,041×
3,257,762×
2,948,466×
2,915,378×
3,055,943×
3,459,591×
3,761,506×
3,131,194×
2,257,940×
1,835,002×
1,882,757×
1,391,739×
1,532,929×
1,365,649×
1,551,231×
1,443,390×
1,655,839×
1,606,671×
1,632,067×
1,812,747×
1,389,087×
1,249,407×
1,381,004×
1,494,604×
1,590,096×
1,901,292×
1,544,993×
1,333,160×
1,090,165×
1,247,403×
901,735×
1,119,561×
1,149,074×
1,020,667×
638,542×
606,361×
826,498×
999,236×
1,065,988×
1,096,447×
790,868×
776,336×
613,612×
1,214,231×
1,346,036×
1,261,572×
1,000,308×
1,191,285×
1,145,148×
1,031,818×
942,238×
1,035,210×
1,125,111×
841,401×
837,878×
881,120×
675,802×
453,319×
294,894×
202,480×
158,809×
107,271×
49,699×
51,277×
109,918×
108,830×
62,620×
3,942×
589×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
0×
1×
66×
0×
0×
0×
0×
0×
0×
736,792×
1,847,920×
4,450,797×
7,620,042×
12,204,873×
17,858,582×
24,605,360×
33,867,849×
45,980,806×
56,702,302×
68,706,101×
83,898,713×
94,851,688×
102,692,290×
108,186,332×
113,279,468×
108,917,087×
98,238,341×
93,648,580×
81,230,516×
72,605,798×
52,221,884×
41,311,691×
29,187,526×
28,117,329×
24,591,214×
22,058,423×
24,953,411×
24,196,853×
23,997,079×
22,699,300×
26,194,150×
24,259,927×
18,451,065×
11,655,480×
10,510,284×
6,413,631×
3,599,242×
1,943,231×
720,218×
119,834×
16,536×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
57s:
58s:
59s:
60s:
61s:
62s:
63s:
64s:
65s:
66s:
67s:
68s:
69s:
70s:
71s:
72s:
73s:
74s:
75s:
76s:
77s:
78s:
79s:
80s:
81s:
82s:
83s:
84s:
85s:
86s:
87s:
88s:
89s:
90s:
91s:
92s:
93s:
94s:
95s:
96s:
97s:
98s:
99s:
100s:
101s:
102s:
103s:
104s:
105s:
106s:
107s:
108s:
109s:
110s:
111s:
112s:
113s:
114s:
115s:
116s:
117s:
118s:
119s:
120s:
121s:
122s:
123s:
124s:
125s:
126s:
127s:
128s:
129s:
130s:
131s:
132s:
133s:
134s:
135s:
136s:
137s:
138s:
139s:
140s:
141s:
142s:
143s:
144s:
145s:
146s:
147s:
148s:
149s:
150s:
151s:
152s:
153s:
154s:
155s:
156s:
157s:
158s:
159s:
160s:
161s:
162s:
163s:
164s:
165s:
166s:
167s:
168s:
169s:
170s:
171s:
172s:
173s:
174s:
175s:
176s:
177s:
178s:
179s:
180s:
181s:
182s:
183s:
184s:
185s:
186s:
187s:
188s:
189s:
190s:
191s:
192s:
193s:
194s:
195s:
196s:
197s:
198s:
199s:
200s:
201s:
202s:
203s:
204s:
205s:
206s:
207s:
208s:
209s:
210s:
211s:
212s:
213s:
0×
1×
66×
0×
0×
144,992×
898,768×
4,584,862×
15,153,355×
9,889,802×
13,700,062×
15,907,916×
21,839,683×
45,420,082×
66,280,295×
59,505,597×
66,453,546×
79,995,786×
104,218,363×
92,438,342×
79,607,129×
75,526,054×
77,282,345×
86,179,030×
85,053,384×
89,316,004×
90,408,706×
104,661,164×
122,912,112×
127,382,491×
144,083,859×
158,050,402×
166,359,778×
180,430,342×
160,616,667×
161,278,215×
171,141,994×
177,423,937×
162,380,492×
155,733,065×
173,109,469×
188,720,066×
192,830,946×
207,273,104×
213,687,668×
214,329,206×
218,553,096×
194,524,214×
171,666,936×
208,715,554×
203,721,195×
190,740,409×
185,330,847×
159,717,125×
151,626,352×
153,900,657×
156,551,257×
160,308,593×
178,351,514×
184,685,328×
141,138,303×
108,666,664×
121,579,874×
131,062,271×
127,188,679×
118,327,487×
109,168,086×
116,950,532×
123,060,612×
106,381,580×
105,373,145×
88,472,382×
83,097,231×
105,045,560×
95,160,836×
91,362,973×
93,429,270×
90,688,523×
92,566,596×
91,807,208×
83,444,962×
72,727,188×
65,383,927×
77,813,778×
87,113,369×
92,398,509×
82,173,387×
83,758,212×
73,186,623×
65,179,867×
72,264,757×
72,314,702×
59,228,111×
52,328,505×
48,353,258×
45,713,741×
50,371,186×
58,985,281×
67,574,028×
76,573,516×
78,701,092×
76,563,138×
75,298,819×
74,172,948×
92,699,860×
92,971,585×
90,059,170×
73,627,911×
57,912,852×
50,265,767×
59,867,098×
56,392,784×
47,757,216×
47,559,281×
43,599,208×
36,251,084×
37,759,141×
37,045,670×
33,463,281×
31,037,062×
26,041,109×
33,941,291×
27,598,042×
39,436,799×
40,744,817×
34,427,052×
32,015,361×
33,745,901×
37,175,712×
47,197,982×
48,836,441×
45,180,214×
62,000,423×
56,131,414×
40,814,980×
52,749,638×
58,753,840×
59,596,432×
53,269,035×
60,481,539×
48,091,174×
51,492,586×
58,807,072×
78,962,600×
84,797,637×
100,314,901×
86,208,746×
76,356,957×
72,037,212×
79,255,702×
85,563,498×
84,266,563×
79,108,512×
76,652,869×
82,396,090×
73,040,347×
68,705,349×
65,845,267×
56,898,233×
52,001,191×
40,197,961×
35,117,621×
50,385,344×
53,348,817×
53,565,447×
59,975,561×
69,455,689×
51,161,012×
52,141,473×
61,145,208×
61,619,056×
40,465,815×
34,542,642×
31,183,023×
27,095,223×
30,199,087×
20,109,786×
13,921,609×
12,099,336×
20,011,447×
18,689,341×
19,270,250×
23,083,279×
19,513,625×
13,934,608×
9,873,684×
9,128,246×
10,055,090×
10,878,406×
13,673,377×
10,519,728×
11,341,943×
9,861,396×
12,918,994×
9,224,855×
11,648,112×
8,155,402×
7,903,620×
8,281,687×
12,033,787×
9,157,598×
5,869,663×
6,785,831×
4,492,505×
6,550,030×
7,440,572×
4,428,869×
2,485,496×
3,035,710×
668,858×
387,501×
284,379×
155,332×
39,292×