Base 112: 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		15.25 step
avg		23.66 step
avg		12.20 step
avg		25.94 step
avg		214.51 step
avg
Step Frequency
Full Cycle
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0×
1×
111×
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0×
1×
111×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
28×
3,108×
3,080×
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0×
1×
3,303×
3,190×
6,378×
6,370×
6,358×
6,342×
6,322×
6,298×
6,270×
6,238×
6,202×
6,162×
6,118×
6,070×
6,018×
5,962×
5,902×
5,838×
5,770×
5,698×
5,622×
5,542×
5,458×
5,370×
5,278×
5,182×
5,082×
4,978×
4,870×
4,758×
4,642×
4,522×
4,398×
4,270×
4,138×
4,002×
3,862×
3,718×
3,570×
3,418×
3,262×
3,102×
2,938×
2,770×
2,598×
2,422×
2,242×
2,058×
1,870×
1,678×
1,482×
1,282×
1,078×
870×
658×
442×
222×
0s:
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0×
1×
111×
0×
5,248×
9,552×
38,976×
155,904×
623,616×
2,494,464×
175,360×
157,946×
302,482×
591,472×
1,119,756×
103,908×
34,400×
67,600×
147,420×
299,272×
45,324×
42,992×
73,296×
146,016×
278,224×
0s:
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0×
1×
111×
9,462×
167,358×
376,238×
306,152×
465,064×
834,018×
1,093,448×
1,597,420×
1,717,714×
2,437,690×
2,085,890×
3,331,012×
2,520,382×
4,416,022×
2,953,858×
5,564,248×
3,634,032×
6,757,564×
4,603,890×
7,998,222×
5,819,296×
8,719,826×
6,786,768×
9,109,468×
6,660,668×
8,516,874×
6,090,190×
8,034,934×
5,548,502×
7,331,088×
5,007,024×
6,094,690×
4,515,742×
4,726,130×
3,462,638×
3,443,224×
2,423,062×
2,002,288×
1,353,136×
912,130×
546,482×
266,544×
107,566×
37,714×
3,708×
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0×
1×
3,303×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
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0×
950,604×
2,999,478×
2,302,408×
2,315,502×
3,019,928×
4,150,765×
4,314,152×
4,635,270×
6,040,594×
6,532,057×
6,754,533×
6,940,906×
5,625,902×
5,370,466×
4,090,078×
5,735,768×
5,244,144×
6,078,585×
6,086,458×
5,669,009×
5,242,143×
5,225,326×
4,667,812×
4,400,408×
4,663,814×
6,163,576×
5,154,131×
6,216,681×
5,046,754×
5,952,386×
5,459,337×
5,626,587×
4,648,270×
4,149,714×
3,941,542×
4,525,932×
4,052,646×
4,253,160×
4,380,474×
3,986,678×
2,118,487×
2,073,876×
1,408,450×
1,454,652×
1,404,334×
1,330,006×
1,004,574×
1,238,142×
1,585,476×
1,381,734×
969,322×
1,116,184×
1,341,492×
1,457,784×
1,595,692×
2,174,934×
2,093,008×
2,837,486×
3,230,898×
2,840,382×
2,646,432×
2,778,912×
3,240,393×
2,885,937×
2,852,440×
3,025,580×
2,743,382×
3,397,370×
4,474,720×
3,962,477×
3,865,944×
3,400,334×
5,610,838×
8,122,472×
7,227,790×
7,691,403×
9,308,052×
12,406,163×
13,413,135×
14,686,962×
16,150,320×
19,203,048×
15,566,429×
16,156,118×
15,464,073×
15,552,149×
14,017,524×
13,420,526×
13,803,184×
13,687,116×
14,493,835×
14,279,100×
16,096,546×
16,648,078×
20,236,457×
15,469,264×
17,679,537×
16,109,268×
17,576,956×
18,836,757×
21,298,369×
22,322,409×
23,337,341×
24,346,752×
22,880,778×
22,456,939×
22,362,185×
20,923,388×
21,279,483×
24,629,954×
21,838,130×
20,728,572×
22,034,686×
21,384,581×
21,351,237×
20,159,342×
19,352,006×
18,187,091×
19,234,266×
18,911,630×
19,689,394×
17,984,664×
19,522,928×
20,754,638×
18,943,149×
21,610,853×
26,100,556×
26,349,125×
22,765,202×
22,623,646×
21,611,169×
22,816,979×
22,495,425×
23,404,945×
22,045,207×
22,607,339×
22,543,914×
22,715,613×
23,126,809×
23,500,024×
21,184,640×
19,854,441×
19,704,957×
22,642,802×
23,632,869×
22,299,701×
22,062,768×
21,847,288×
18,410,488×
17,025,539×
19,194,915×
18,156,873×
18,485,776×
18,419,379×
17,368,390×
17,129,952×
17,117,011×
14,404,935×
15,903,749×
13,287,708×
14,228,786×
15,171,832×
13,552,771×
15,229,130×
14,829,912×
12,024,966×
12,536,979×
12,701,891×
12,519,622×
11,985,285×
9,838,134×
8,409,262×
7,887,497×
7,652,615×
4,544,038×
4,296,770×
4,045,360×
4,426,648×
3,495,946×
4,233,330×
5,057,406×
5,352,292×
6,131,839×
6,945,046×
7,321,212×
8,689,251×
8,452,908×
9,535,924×
7,738,266×
8,853,806×
10,197,667×
8,736,860×
9,611,288×
10,618,928×
11,773,844×
11,307,118×
10,452,429×
10,076,183×
11,038,500×
11,319,653×
10,724,964×
10,878,811×
10,210,077×
9,025,745×
9,865,164×
9,282,700×
9,131,878×
8,043,095×
8,553,290×
9,260,207×
9,924,650×
10,004,286×
10,672,840×
11,086,352×
11,353,036×
12,348,333×
15,988,698×
15,078,546×
16,290,843×
14,960,466×
13,749,711×
11,489,504×
11,231,548×
11,004,998×
11,386,207×
11,454,303×
11,884,833×
8,997,960×
9,416,309×
9,116,611×
8,060,482×
7,635,979×
8,160,898×
6,964,182×
6,980,288×
6,205,277×
5,467,289×
6,098,088×
5,638,036×
5,764,352×
5,413,296×
5,040,556×
5,546,384×
5,577,076×
7,531,146×
8,505,970×
8,511,401×
9,669,293×
9,390,750×
10,565,198×
10,811,242×
11,711,679×
12,346,715×
13,480,542×
10,852,928×
12,354,153×
12,549,143×
11,930,752×
11,134,502×
9,276,601×
7,212,569×
8,651,343×
7,699,110×
10,739,581×
11,622,447×
11,451,024×
9,353,924×
9,983,083×
8,234,635×
9,261,760×
9,284,228×
9,015,249×
8,276,382×
8,950,497×
5,794,883×
6,292,418×
4,844,168×
4,262,283×
3,275,010×
3,832,582×
3,535,914×
4,306,172×
3,334,162×
3,579,810×
5,282,871×
5,019,854×
5,023,795×
5,145,196×
5,215,974×
5,335,135×
4,735,085×
5,941,122×
4,935,390×
2,811,830×
2,626,696×
2,652,202×
1,508,902×
1,337,711×
1,070,932×
788,064×
756,230×
781,890×
886,580×
1,180,639×
1,087,672×
1,248,896×
786,094×
1,002,096×
762,526×
500,180×
442,220×
563,966×
640,738×
580,732×
554,232×
448,388×
313,184×
554,780×
714,440×
467,848×
452,568×
186,724×
220,928×