Base 108: Iteration Frequencies: Full Cycle

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
Full Cycle		1.99 step
avg		19.17 step
avg		22.91 step
avg		12.71 step
avg		24.40 step
avg		233.26 step
avg
Step Frequency
Full Cycle
0s:
1s:
2s:
0×
1×
107×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
0×
1×
107×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
27×
2,889×
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:
56s:
0×
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:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
0×
1×
107×
0×
8,880×
17,928×
72,576×
290,304×
0×
0×
81,392×
217,632×
870,912×
3,483,648×
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:
46s:
0×
1×
49,391×
73,172×
116,216×
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:
346s:
347s:
348s:
349s:
350s:
351s:
352s:
353s:
354s:
355s:
356s:
357s:
358s:
359s:
360s:
361s:
362s:
363s:
364s:
365s:
366s:
367s:
368s:
369s:
370s:
371s:
372s:
373s:
374s:
375s:
376s:
377s:
378s:
379s:
380s:
381s:
382s:
383s:
384s:
385s:
386s:
387s:
388s:
389s:
390s:
391s:
392s:
393s:
394s:
395s:
396s:
397s:
398s:
399s:
400s:
401s:
402s:
403s:
404s:
405s:
406s:
407s:
408s:
409s:
410s:
411s:
412s:
413s:
414s:
415s:
416s:
417s:
418s:
419s:
420s:
421s:
422s:
423s:
424s:
425s:
426s:
427s:
428s:
429s:
430s:
431s:
432s:
433s:
434s:
435s:
436s:
437s:
438s:
439s:
440s:
441s:
442s:
443s:
444s:
445s:
446s:
447s:
448s:
449s:
450s:
451s:
452s:
453s:
454s:
455s:
456s:
457s:
458s:
459s:
460s:
461s:
462s:
463s:
464s:
465s:
466s:
467s:
468s:
469s:
470s:
471s:
472s:
473s:
474s:
475s:
476s:
477s:
478s:
479s:
480s:
481s:
482s:
483s:
484s:
485s:
486s:
487s:
488s:
489s:
490s:
491s:
492s:
493s:
494s:
495s:
496s:
497s:
498s:
499s:
500s:
501s:
502s:
503s:
504s:
505s:
506s:
507s:
508s:
509s:
510s:
511s:
512s:
513s:
514s:
515s:
516s:
517s:
518s:
519s:
520s:
521s:
0×
1×
3,077×
0×
0×
56,720×
161,836×
110,490×
137,859×
285,657×
245,002×
373,120×
358,652×
565,768×
788,951×
714,756×
716,326×
993,324×
1,177,702×
1,530,706×
2,485,610×
2,204,414×
1,971,120×
1,669,800×
1,999,080×
2,435,870×
2,809,240×
3,526,587×
3,143,966×
2,922,994×
3,549,601×
3,329,012×
3,771,656×
3,639,723×
3,942,604×
4,511,308×
4,950,476×
4,479,446×
3,996,829×
4,659,239×
5,724,634×
4,926,311×
5,342,745×
5,509,828×
6,829,152×
8,021,972×
8,195,779×
6,812,766×
7,040,335×
6,803,278×
5,973,278×
6,162,895×
7,131,373×
7,753,088×
7,661,551×
7,942,633×
9,548,405×
9,649,969×
9,338,578×
8,589,304×
9,335,395×
9,278,674×
8,752,301×
8,325,118×
9,812,208×
9,904,021×
9,094,663×
8,138,228×
7,128,942×
7,096,701×
7,645,973×
6,968,470×
6,728,366×
6,599,419×
7,299,557×
6,565,471×
7,366,519×
7,424,334×
7,255,640×
7,762,107×
7,227,072×
7,831,564×
8,253,178×
7,595,134×
7,497,339×
7,952,701×
7,963,005×
7,151,816×
7,650,168×
7,892,314×
7,785,656×
7,505,157×
6,527,751×
7,070,979×
6,559,918×
7,150,422×
6,383,349×
6,126,238×
5,210,194×
5,037,271×
4,565,208×
5,078,568×
4,785,986×
5,025,301×
3,509,868×
3,571,940×
3,047,674×
3,455,484×
3,320,305×
4,235,053×
3,838,479×
3,468,551×
3,698,002×
3,893,387×
3,878,349×
3,170,698×
3,082,343×
3,848,225×
3,946,573×
4,048,674×
3,877,068×
3,580,016×
2,929,616×
3,088,845×
2,781,840×
2,023,324×
2,667,740×
3,212,526×
3,241,232×
3,828,432×
3,439,457×
2,867,234×
2,550,495×
2,536,502×
2,137,785×
2,123,772×
2,383,788×
1,820,430×
2,034,988×
2,299,482×
2,057,992×
2,283,966×
2,439,977×
2,496,624×
2,753,582×
2,396,895×
2,323,192×
2,453,191×
2,070,164×
2,892,685×
2,990,093×
3,459,466×
3,425,026×
3,377,325×
3,689,819×
3,365,929×
3,170,351×
2,962,745×
3,417,408×
3,936,613×
4,668,382×
4,835,658×
4,771,967×
4,780,721×
4,687,554×
4,768,405×
4,960,257×
5,484,056×
5,535,342×
5,961,876×
5,366,864×
6,034,873×
5,657,188×
6,760,157×
7,553,767×
7,167,864×
7,528,002×
11,118,081×
14,029,988×
13,340,140×
14,806,280×
16,490,483×
18,097,127×
16,496,294×
14,886,787×
14,948,611×
14,105,504×
14,060,028×
12,080,092×
11,064,646×
9,900,636×
9,382,796×
8,309,405×
8,590,484×
8,005,247×
8,807,309×
8,667,442×
9,004,574×
9,078,237×
7,905,045×
8,083,681×
6,589,722×
7,087,901×
7,340,534×
7,370,997×
6,759,979×
6,419,248×
6,247,762×
6,989,674×
6,700,120×
7,436,344×
6,719,916×
8,004,586×
8,054,771×
7,164,698×
7,041,000×
7,114,343×
7,562,634×
7,562,496×
7,723,320×
8,943,927×
10,096,070×
9,921,487×
8,719,982×
7,955,619×
7,448,936×
8,229,485×
8,257,071×
9,199,023×
8,713,381×
9,322,744×
9,856,761×
9,680,187×
8,859,064×
8,547,252×
8,554,378×
8,130,322×
8,307,857×
7,165,267×
6,990,813×
7,053,046×
7,649,261×
9,054,471×
9,793,957×
8,622,059×
8,220,028×
8,530,917×
8,304,183×
7,327,792×
7,995,677×
8,756,173×
8,893,622×
8,443,976×
8,151,719×
7,917,118×
7,017,406×
6,487,818×
6,974,633×
7,532,562×
6,690,425×
7,729,090×
8,024,197×
7,392,821×
5,983,801×
5,995,730×
5,467,592×
4,952,070×
5,014,168×
5,263,581×
5,749,099×
5,493,344×
5,737,427×
4,759,939×
4,484,054×
4,474,941×
3,994,315×
4,547,753×
4,780,530×
4,753,728×
3,585,962×
3,891,174×
4,766,136×
3,711,158×
3,650,782×
2,742,723×
2,750,408×
2,706,350×
3,644,436×
3,576,336×
3,375,480×
3,595,476×
3,003,496×
3,281,720×
4,249,351×
4,227,577×
4,424,774×
4,549,360×
4,206,561×
3,373,445×
3,377,878×
3,596,805×
3,422,790×
3,320,352×
3,548,322×
3,424,949×
4,029,953×
4,319,837×
4,523,974×
4,268,896×
4,186,324×
3,970,072×
4,119,758×
4,718,446×
5,229,057×
3,911,997×
3,985,768×
4,601,451×
4,458,120×
4,241,819×
3,629,661×
3,124,906×
2,861,088×
2,718,046×
2,729,124×
2,384,230×
2,832,965×
3,329,768×
2,969,022×
3,368,951×
3,409,253×
3,648,893×
3,548,237×
3,622,082×
3,516,207×
2,916,997×
2,844,862×
2,860,806×
3,006,130×
3,656,222×
3,645,164×
3,639,013×
3,893,461×
4,228,432×
3,426,889×
4,382,665×
4,567,316×
4,299,014×
4,491,424×
4,654,700×
5,156,690×
5,349,662×
5,698,210×
5,766,643×
5,581,306×
5,101,414×
5,607,990×
6,126,116×
5,311,759×
5,744,835×
6,266,423×
5,806,027×
6,683,798×
6,536,836×
5,588,277×
6,072,909×
5,355,968×
5,825,803×
4,906,130×
5,300,895×
5,267,389×
5,123,875×
6,015,011×
5,845,120×
5,526,637×
5,857,703×
6,093,114×
6,799,342×
5,893,916×
5,935,053×
5,648,725×
6,239,734×
6,022,518×
5,970,269×
7,160,117×
7,191,215×
8,209,935×
7,743,513×
8,642,220×
7,675,959×
7,941,721×
7,704,747×
7,623,787×
6,783,449×
6,128,175×
5,865,221×
5,909,922×
6,092,203×
6,240,652×
6,264,252×
6,060,163×
6,167,843×
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×