Scrutinizing Kaprekar's Constant



What the 6174 Scrutinizer™ algorithm does...

  1. Re-arrange digits from highest to lowest.
  2. Subtract the reverse value.
  3. Repeat the process using the resulting difference until a “Fixed Point” or recurring “mathematical loop” occurs.

So what's all this about Kaprekar's Constant and 6174...?

So if you don't know, Numberphile posted a video about the number 6174 on Youtube back in December of 2011. The scholarly gentleman claims some unique properties about the number that, on the surface seem plausible. However, after further investigation, it can be proven that
* the claims in the video are demonstrably false.

Kaprekar's Constant: 6174
1) Start with any four-digit number, using at least two different digits.(Leading zeros are allowed.) example... 2047
2) Re-arrange the digits from highest to lowest 7420
3) Subtract the reverse value 7420 - 0247 = 7173
4) Repeat the process using the resulting difference until a Fixed Point (recurring mathematical loop) occurs, or the algorithm resolves to zero 7731 - 1377 = 6354
6543 - 3456 = 3087
8730 - 0378 = 8352
8532 - 2358 = 6174
7641 - 1467 = 6174
When starting with a four-digit number having at least 2 unique digits, Kaprekar's constant states that the algorithm will always find it's Fixed Point at 6174 within 8 iterations.

Where it goes off the rails...

Regarding four-digit numbers and Kaprekar's Constant (6174) Wikipedia's definition of Kaprekar's Constant ... 999 also turns up as an end-result, if the 4-digit number sticks to a pattern like.. 1000, 1112, 2221, 3334, 4443, 5556, 6665, 7776... etc and their permutations. This is also true for 3-digit patterns like 112, 223, 334, 445, 455, 566, 677 ... etc (*where 495 ∝ 6174 & 99 ∝ 999)

*It can be noted that the only true constant that can be derived from this algorithm is "9"... the result from 2-digit number sequences where the digits are not the same.

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Programmed by Dan Araquel 09.2018