Base 108: Iteration Frequencies: Cycle Start

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDEF
GHIJKLMNOPQRSTUV
WXYZabcdefghijkl
mnopqrstuvwxyzΑΒ
ΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣ
ΤΥΦΧΨΩαβγδεζηθικ
λμνξοπρςστυφχψωА

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
Average Steps
Cycle Start		0.99 step
avg		1.48 step
avg		21.91 step
avg		4.10 step
avg		23.40 step
avg		117.74 step
avg
Step Frequency
Cycle Start
0s:
1s:
1×
107×
0s:
1s:
2s:
28×
2,996×
2,862×
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:
1×
3,077×
2,968×
5,934×
5,926×
5,914×
5,898×
5,878×
5,854×
5,826×
5,794×
5,758×
5,718×
5,674×
5,626×
5,574×
5,518×
5,458×
5,394×
5,326×
5,254×
5,178×
5,098×
5,014×
4,926×
4,834×
4,738×
4,638×
4,534×
4,426×
4,314×
4,198×
4,078×
3,954×
3,826×
3,694×
3,558×
3,418×
3,274×
3,126×
2,974×
2,818×
2,658×
2,494×
2,326×
2,154×
1,978×
1,798×
1,614×
1,426×
1,234×
1,038×
838×
634×
426×
214×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
1×
90,379×
235,560×
943,488×
3,773,952×
245,488×
219,973×
405,460×
24,372×
17,760×
32,572×
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:
1×
58,191×
67,342×
113,246×
243,874×
407,844×
698,470×
894,028×
1,290,384×
1,223,986×
2,216,280×
2,133,812×
3,182,770×
2,533,470×
4,175,514×
2,964,222×
5,209,338×
3,773,284×
6,307,622×
4,854,822×
7,072,408×
5,250,406×
7,195,582×
5,494,040×
7,184,414×
5,379,228×
6,939,028×
4,963,960×
5,973,546×
4,553,710×
5,323,248×
3,912,136×
4,308,742×
3,390,476×
3,301,708×
2,677,786×
2,421,510×
1,885,444×
1,541,782×
1,155,390×
861,490×
574,580×
298,616×
116,386×
26,768×
2,828×
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:
214s:
215s:
216s:
217s:
218s:
219s:
220s:
221s:
222s:
223s:
224s:
225s:
226s:
227s:
228s:
229s:
230s:
231s:
232s:
233s:
234s:
235s:
236s:
237s:
238s:
239s:
240s:
241s:
242s:
243s:
244s:
245s:
246s:
247s:
248s:
249s:
250s:
251s:
252s:
253s:
254s:
255s:
256s:
257s:
258s:
259s:
260s:
261s:
262s:
263s:
264s:
265s:
266s:
267s:
268s:
269s:
270s:
271s:
272s:
273s:
274s:
275s:
276s:
277s:
278s:
279s:
280s:
281s:
282s:
283s:
284s:
285s:
286s:
287s:
288s:
289s:
290s:
291s:
292s:
293s:
294s:
295s:
296s:
297s:
298s:
299s:
300s:
301s:
302s:
303s:
304s:
305s:
306s:
307s:
308s:
309s:
310s:
311s:
312s:
313s:
314s:
315s:
316s:
317s:
318s:
319s:
320s:
321s:
322s:
323s:
324s:
325s:
326s:
327s:
328s:
329s:
330s:
331s:
332s:
333s:
334s:
335s:
336s:
337s:
338s:
339s:
340s:
341s:
342s:
343s:
344s:
345s:
1×
3,856,789×
10,040,870×
7,998,760×
9,453,015×
11,986,171×
14,894,886×
13,820,470×
13,254,173×
13,088,020×
12,845,557×
14,149,044×
12,464,486×
12,627,154×
13,187,327×
12,145,581×
10,825,042×
12,293,001×
13,758,802×
14,618,019×
15,093,378×
15,226,044×
15,319,571×
15,455,304×
14,795,049×
14,097,732×
14,695,975×
15,731,873×
14,742,226×
14,310,947×
13,216,445×
12,459,750×
13,299,138×
12,987,261×
12,700,550×
12,110,232×
13,527,660×
12,995,650×
12,339,141×
12,265,464×
11,107,181×
11,631,753×
11,902,464×
12,328,528×
12,732,928×
14,280,476×
14,889,594×
14,280,754×
14,297,637×
14,489,798×
15,413,084×
16,069,495×
16,881,785×
15,875,250×
16,595,222×
16,307,124×
15,421,957×
15,007,087×
14,091,118×
13,238,704×
12,549,148×
11,998,367×
11,280,238×
11,933,158×
12,089,642×
12,852,385×
13,755,426×
15,025,664×
13,727,366×
13,633,735×
13,987,603×
13,861,845×
12,800,972×
13,264,473×
14,195,556×
14,977,792×
14,055,854×
13,708,557×
13,494,028×
12,153,338×
11,312,922×
12,284,427×
12,931,036×
12,345,138×
13,027,657×
12,443,278×
12,416,290×
10,150,935×
10,642,059×
9,702,031×
9,253,796×
8,853,346×
9,106,966×
8,753,797×
8,904,680×
9,157,981×
8,457,275×
7,414,146×
7,526,291×
6,633,995×
7,631,987×
7,854,527×
8,803,881×
7,223,931×
7,129,089×
8,150,864×
7,299,759×
7,292,127×
5,878,085×
5,755,543×
6,486,983×
7,393,581×
7,241,556×
7,046,954×
6,995,109×
5,853,938×
6,285,527×
7,031,191×
6,250,901×
7,092,514×
7,761,886×
7,447,793×
7,201,877×
6,817,335×
6,464,039×
5,973,285×
5,856,854×
5,686,107×
5,548,721×
6,413,741×
6,140,267×
6,558,962×
6,568,378×
6,244,316×
6,254,038×
6,559,735×
7,215,070×
7,982,639×
6,308,892×
6,308,960×
7,054,642×
6,528,284×
7,134,504×
6,619,754×
6,584,372×
6,286,114×
6,095,371×
6,418,943×
5,750,159×
6,003,316×
6,292,513×
6,386,430×
7,305,564×
8,077,635×
8,484,551×
8,320,204×
8,402,803×
8,203,761×
7,685,402×
7,805,119×
8,344,862×
8,541,472×
9,618,098×
9,012,028×
9,673,886×
9,550,649×
10,988,589×
10,980,656×
11,550,529×
12,095,318×
13,114,847×
12,688,874×
12,867,030×
13,503,357×
13,255,213×
12,693,437×
12,786,185×
11,619,729×
10,953,703×
11,370,294×
11,074,018×
9,734,734×
9,810,225×
9,485,877×
8,939,585×
9,200,318×
8,777,362×
7,098,419×
7,794,903×
6,999,861×
7,526,189×
6,607,783×
6,880,533×
6,685,005×
6,189,051×
7,234,427×
7,231,478×
7,215,778×
7,476,733×
7,175,090×
7,690,487×
6,759,742×
6,796,355×
6,649,931×
7,237,834×
7,247,284×
6,964,513×
8,091,361×
8,044,679×
9,197,431×
9,065,203×
9,853,662×
8,444,813×
8,828,769×
8,594,923×
8,213,223×
7,268,551×
6,723,257×
6,183,107×
6,230,948×
6,356,183×
6,409,664×
6,439,212×
6,132,611×
6,276,607×
5,244,151×
5,942,279×
5,178,072×
4,907,004×
5,485,131×
4,665,808×
6,291,462×
5,924,635×
6,062,055×
6,129,551×
7,010,034×
7,164,461×
6,459,552×
7,399,311×
6,979,365×
7,413,718×
5,721,862×
5,834,321×
5,881,899×
5,523,326×
4,922,478×
4,329,287×
3,858,149×
3,514,493×
2,789,664×
2,480,556×
2,291,192×
2,474,169×
2,122,562×
1,879,040×
1,321,003×
1,941,230×
1,406,871×
1,034,130×
1,327,389×
918,746×
1,157,332×
1,888,527×
1,750,691×
1,971,851×
2,157,295×
1,982,624×
1,843,983×
1,318,470×
1,633,478×
1,902,529×
1,801,917×
1,717,960×
1,842,280×
2,201,858×
2,055,958×
1,879,019×
1,837,744×
1,287,025×
921,714×
1,174,368×
1,040,586×
911,946×
878,544×
1,178,628×
1,919,682×
2,289,408×
1,700,205×
1,488,300×
2,406,156×
1,792,897×
2,098,269×
2,240,754×
2,021,042×
2,065,643×
1,833,096×
1,901,304×
1,858,704×
1,659,912×
2,368,744×
2,498,415×
2,159,877×
2,482,882×
2,726,704×
3,434,781×
3,438,380×
2,883,770×
3,106,877×
2,503,695×
1,826,247×
1,940,630×
1,855,194×
1,713,558×
1,382,662×
1,383,047×
1,465,706×
1,298,024×
1,060,762×
789,487×
823,240×
1,189,733×
741,279×
866,252×
737,096×
584,512×
608,418×
638,160×
973,119×
806,252×
881,860×
491,632×
649,575×
647,836×
448,649×
677,104×
379,380×
626,304×
470,512×
873,642×
1,263,096×
571,614×
472,816×
293,266×
143,952×
174,824×
88,808×