Base 101: 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		8.50 step
avg		22.09 step
avg		51.27 step
avg		23.79 step
avg		214.72 step
avg
Step Frequency
Full Cycle
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100×
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0×
2×
166×
83×
104×
420×
376×
800×
8×
776×
816×
1,600×
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0×
1×
100×
5,201×
2,597×
5,192×
5,182×
5,168×
5,150×
5,128×
5,102×
5,072×
5,038×
5,000×
4,958×
4,912×
4,862×
4,808×
4,750×
4,688×
4,622×
4,552×
4,478×
4,400×
4,318×
4,232×
4,142×
4,048×
3,950×
3,848×
3,742×
3,632×
3,518×
3,400×
3,278×
3,152×
3,022×
2,888×
2,750×
2,608×
2,462×
2,312×
2,158×
2,000×
1,838×
1,672×
1,502×
1,328×
1,150×
968×
782×
592×
398×
200×
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0×
1×
100×
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0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
22,350×
72,025×
561×
0×
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0×
0×
0×
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0×
0×
0×
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0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
1,102,200×
3,263,800×
134,489×
2,600×
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0×
1×
100×
0×
0×
161,704×
175,566×
237,272×
523,988×
736,240×
1,147,034×
1,265,143×
1,857,152×
1,602,247×
2,788,157×
2,291,142×
3,629,150×
2,645,060×
4,490,183×
3,482,009×
5,415,978×
4,580,334×
5,875,456×
4,610,443×
5,445,490×
4,239,791×
5,240,635×
3,813,080×
4,685,641×
3,478,441×
3,773,936×
3,115,347×
3,035,128×
2,523,712×
2,236,865×
1,947,970×
1,600,561×
1,309,029×
934,566×
633,514×
421,044×
335,138×
206,903×
69,496×
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0×
1×
100×
73,040×
196,755×
864,001×
1,395,227×
4,206,723×
2,020,957×
907,289×
1,343,482×
1,941,236×
2,348,250×
1,768,019×
2,314,364×
2,644,759×
3,024,272×
2,790,660×
3,128,602×
3,805,633×
3,806,057×
4,099,127×
3,603,708×
4,216,429×
4,813,148×
4,605,681×
4,423,598×
3,937,751×
3,377,964×
3,351,324×
3,349,873×
2,922,919×
2,823,894×
2,895,108×
2,571,832×
3,201,673×
3,326,920×
3,034,162×
2,228,197×
2,506,906×
3,308,435×
3,633,670×
3,336,339×
3,808,300×
3,499,333×
3,765,984×
3,182,849×
3,045,695×
2,372,175×
2,185,407×
2,057,795×
2,105,283×
2,134,478×
2,774,584×
2,784,358×
3,659,013×
4,580,196×
7,215,004×
6,591,535×
6,946,652×
6,688,536×
6,127,116×
6,138,217×
7,002,027×
6,167,826×
6,445,292×
6,659,051×
6,810,348×
7,278,774×
7,884,998×
7,288,436×
6,990,831×
7,364,844×
7,528,137×
6,678,427×
5,572,397×
4,484,997×
5,083,147×
6,099,296×
5,811,976×
6,496,602×
6,237,100×
5,634,268×
5,901,316×
6,411,979×
7,589,141×
10,075,983×
9,308,123×
9,098,366×
8,774,596×
8,267,133×
7,800,765×
9,314,931×
10,751,921×
11,108,323×
11,315,106×
12,472,267×
11,888,347×
12,006,876×
11,796,641×
11,175,202×
10,439,226×
10,320,434×
8,425,586×
8,367,966×
8,921,180×
7,906,597×
5,508,418×
5,324,231×
6,583,913×
5,914,284×
4,650,686×
3,881,747×
4,543,586×
5,046,840×
5,872,161×
6,447,095×
5,657,014×
5,229,854×
4,614,173×
5,422,677×
5,989,304×
7,095,551×
6,208,806×
6,194,552×
4,454,824×
4,340,099×
3,962,295×
3,719,459×
3,654,335×
2,799,456×
2,600,231×
2,767,480×
2,335,617×
1,962,694×
1,887,515×
1,967,062×
2,540,222×
2,913,120×
3,025,973×
3,120,384×
3,273,016×
3,695,821×
4,421,510×
4,711,519×
4,912,956×
4,706,029×
2,878,702×
3,393,769×
3,629,735×
3,596,861×
3,073,570×
3,621,276×
3,793,424×
3,716,328×
3,596,075×
3,344,070×
3,034,381×
3,058,076×
2,977,061×
2,710,297×
3,072,850×
3,636,449×
4,097,520×
4,114,720×
3,545,705×
3,118,390×
2,361,061×
2,560,754×
2,320,962×
2,603,566×
2,929,268×
2,936,509×
3,434,755×
3,706,477×
3,734,419×
4,251,947×
4,521,823×
3,662,671×
3,184,334×
3,944,006×
3,625,402×
2,917,660×
2,942,256×
3,093,310×
2,841,148×
2,830,165×
2,165,423×
2,141,040×
1,994,904×
2,439,339×
3,156,631×
1,510,668×
1,630,927×
1,212,739×
1,130,765×
1,457,528×
2,081,376×
2,755,758×
3,356,269×
2,263,897×
2,079,331×
1,794,631×
1,875,802×
2,576,853×
2,764,834×
1,777,824×
1,353,544×
1,346,943×
1,407,017×
2,118,676×
1,988,709×
1,602,192×
2,233,652×
2,662,948×
3,271,232×
2,196,179×
1,817,611×
2,051,262×
1,860,449×
2,227,620×
2,496,834×
1,918,855×
1,492,831×
1,451,111×
1,364,583×
1,536,355×
1,373,619×
1,275,874×
1,494,452×
1,827,533×
1,304,971×
1,317,195×
1,645,185×
1,711,204×
1,657,943×
1,935,155×
2,813,007×
2,348,248×
2,018,075×
2,455,147×
2,844,772×
3,239,948×
4,254,838×
4,684,592×
4,209,356×
4,964,161×
5,348,251×
6,468,303×
5,606,102×
6,199,005×
6,854,736×
6,363,142×
5,102,045×
4,086,504×
4,275,273×
5,744,718×
5,552,271×
4,603,810×
4,446,844×
4,537,700×
4,145,049×
3,395,901×
2,989,101×
3,397,572×
3,755,952×
3,881,123×
3,615,881×
4,434,310×
3,536,650×
3,299,928×
2,624,650×
2,237,422×
2,494,584×
2,636,893×
2,676,750×
2,867,378×
3,892,633×
3,976,548×
4,017,932×
3,780,426×
3,844,084×
3,921,545×
3,891,421×
4,325,637×
4,594,650×
5,204,097×
4,218,904×
4,983,049×
5,780,807×
5,885,590×
4,659,995×
4,834,514×
4,338,088×
4,944,069×
4,774,344×
4,613,431×
3,808,435×
4,265,598×
4,015,185×
3,882,349×
3,232,515×
3,099,046×
3,678,558×
3,998,740×
2,356,248×
2,142,942×
1,991,506×
1,139,616×
1,146,627×
1,158,126×
1,476,017×
1,898,058×
1,590,119×
1,521,559×
1,760,206×
1,803,929×
1,637,469×
1,356,669×
1,440,614×
1,402,022×
1,512,506×
1,447,569×
1,759,453×
1,464,910×
1,891,814×
1,844,580×
3,081,664×
2,227,095×
2,172,277×
1,092,110×
1,671,266×
1,955,523×
2,329,794×
2,545,898×
2,163,126×
2,418,124×
1,463,281×
1,323,937×
1,552,499×
1,874,100×
1,930,193×
1,776,199×
1,641,286×
1,938,183×
2,623,828×
2,282,334×
1,896,685×
1,179,503×
835,064×
1,100,738×
1,584,548×
1,810,506×
2,186,343×
2,032,590×
2,113,517×
2,314,377×
2,299,215×
1,824,510×
2,075,955×
2,991,565×
2,371,121×
2,417,883×
1,818,196×
2,049,931×
1,952,367×
2,081,778×
2,061,979×
1,892,893×
1,936,211×
1,931,059×
1,918,332×
1,417,303×
1,249,391×
1,521,231×
1,895,138×
1,835,641×
1,705,541×
2,840,309×
2,314,926×
1,771,615×
1,721,281×
1,814,562×
2,024,673×
2,259,743×
2,457,758×
1,933,850×
2,174,681×
2,394,819×
3,280,180×
3,449,584×
2,736,586×
2,776,364×
2,487,501×
3,448,527×
3,298,830×
4,160,743×
3,972,990×
5,025,141×
5,374,620×
5,184,047×
5,506,833×
5,891,973×
3,944,794×
3,940,945×
4,575,880×
4,232,376×
4,302,143×
4,259,574×
4,928,724×
5,464,894×
5,948,084×
5,950,074×
6,147,076×
6,920,562×
6,053,660×
6,375,166×
6,755,459×
6,058,363×
6,273,839×
5,775,726×
4,847,837×
3,344,447×
2,993,501×
3,117,908×
3,438,071×
3,296,133×
3,369,922×
3,232,699×
3,025,034×
3,100,863×
2,962,069×
2,514,940×
2,784,760×
2,932,744×
2,259,121×
1,895,700×
1,737,654×
1,728,011×
1,758,021×
2,020,679×
2,076,675×
2,156,998×
2,568,761×
2,496,177×
2,326,829×
1,685,556×
1,477,035×
1,583,176×
1,864,751×
1,994,130×
1,905,044×
1,934,708×
1,669,374×
2,150,985×
2,850,942×
2,381,209×
2,093,680×
1,868,943×
2,447,118×
3,288,014×
3,468,235×
2,016,290×
1,829,364×
2,221,351×
2,394,427×
2,071,312×
2,108,903×
1,728,972×
1,210,924×
1,611,635×
1,899,755×
862,065×
735,350×
1,120,534×
412,758×
404,424×
775,024×
1,004,039×
909,393×
1,066,224×
1,349,482×
654,788×
989,389×
1,342,762×
1,375,286×
1,355,195×
1,766,063×
1,258,151×
1,714,209×
1,629,117×
1,415,284×
1,269,203×
1,312,050×
1,967,327×
923,096×
688,263×
862,153×
941,415×
821,285×
864,577×
648,138×
337,331×
174,358×
92,123×
44,926×
22,512×
15,991×
7,122×