350:
in at most 6 iterations. Sometimes these numbers (495, 6174, and their counterparts in other digit lengths or in bases other than 10) are called "Peyush constants" named after Peyush Dixit who solved this routine as a part of his IMO 2000 (International
Mathematical Olympiad, Year 2000) thesis.
100:
after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4. For numbers with three identical digits and a fourth digit that is one higher or lower (such as 2111), it is essential to treat 3-digit numbers with a leading
346:
There can be analogous fixed points for digit lengths other than four; for instance, if we use 3-digit numbers, then most sequences (i.e., other than repdigits such as 111) will terminate in the value
70:, 6174, in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641 โ 1467 = 6174. For example, choose 1459:
567:
52:
Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.
547:
562:
101:
zero; for example: 2111 โ 1112 = 0999; 9990 โ 999 = 8991; 9981 โ 1899 = 8082; 8820 โ 288 = 8532; 8532 โ 2358 = 6174.
67:
526:
269:
500:
464:
341:
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1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 343, 441, 686, 882, 1029, 2058, 3087, 6174
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Take any four-digit number, using at least two different digits (leading zeros are allowed).
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Sample (C) code to walk the first 10000 numbers and their steps to
Kaprekar's Constant
556:
410:
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The only four-digit numbers for which
Kaprekar's routine does not reach 6174 are
17:
530:
321:
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18 + 18 + 18 = 5832 + 324 + 18 = 6174, and coincidentally, 6 + 1 + 7 + 4 = 18.
308:
97:
542:
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Sample (Python) code to walk any four-digit number to
Kaprekar's Constant
93:
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Sample (Perl) code to walk any four-digit number to
Kaprekar's Constant
219:
138:
282:
179:
425:
Kaprekar DR (1955). "An
Interesting Property of the Number 6174".
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36:
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6174 can be written as the sum of the first three powers of 18:
378:
The sum of squares of the prime factors of 6174 is a square:
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2 + 3 + 3 + 7 + 7 + 7 = 4 + 9 + 9 + 49 + 49 + 49 = 169 = 13
149:
367:, i.e. none of its prime factors are greater than 7.
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Kaprekar's routine ยง Definition and properties
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Subtract the smaller number from the bigger number.
45:. This number is renowned for the following rule:
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446:Kaprekar DR (1980). "On Kaprekar Numbers".
204:(six thousand one hundred seventy-fourth)
393:
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192:six thousand one hundred seventy-four
7:
96:such as 1111, which give the result
478:"Kaprekar's Iterations and Numbers"
448:Journal of Recreational Mathematics
25:
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1:
519:"6174 is Kaprekar's Constant"
58:Go back to step 2 and repeat.
336:Other "Kaprekar's constants"
62:The above process, known as
589:
339:
128:
527:University of Nottingham
406:"Mysterious number 6174"
66:, will always reach its
568:Mathematical constants
503:, p. 14, Operations.
482:www.cut-the-knot.org
563:Arithmetic dynamics
428:Scripta Mathematica
33:Kaprekar's constant
81:8532 โ 2358 = 6174
78:8820 โ 0288 = 8532
75:9541 โ 1459 = 8082
64:Kaprekar's routine
467:, p. 1, Overview.
402:Nishiyama, Yutaka
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18:Kaprekar constant
16:(Redirected from
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358:Other properties
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134:List of numbers
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404:(March 2006).
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104:Natural number
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84:7641 โ 1467 =
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43:D. R. Kaprekar
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411:Plus Magazine
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365:smooth number
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240:Roman numeral
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230:Greek numeral
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501:Hanover 2017
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485:. Retrieved
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465:Hanover 2017
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363:6174 is a 7-
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247:MCLXXIV, or
120:6175 →
114:← 6173
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31:is known as
28:
26:
531:Brady Haran
523:Numberphile
454:(2): 81โ82.
322:Hexadecimal
68:fixed point
27:The number
557:Categories
487:2022-09-21
435:: 244โ245.
388:References
309:Duodecimal
35:after the
214:2 ร 3 ร 7
94:repdigits
573:Integers
274:22110200
220:Divisors
188:Cardinal
139:Integers
270:Ternary
198:Ordinal
283:Senary
257:Binary
251:CLXXIV
234:,ฯฮกฮฮยด
202:6174th
37:Indian
300:14036
296:Octal
287:44330
326:181E
313:36A6
117:6174
98:0000
86:6174
29:6174
348:495
559::
529::
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521:.
480:.
452:13
450:.
433:15
431:.
408:.
328:16
315:12
249:VI
177:9k
174:8k
171:7k
168:6k
165:5k
162:4k
159:3k
156:2k
153:1k
533:.
490:.
414:.
302:8
289:6
276:3
263:2
245:V
180:โ
150:0
147:โ
20:)
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