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Kaprekar's Magic 6174
Kaprekar’s routine is a simple arithmetic procedure which, when applied to four digit numbers, rapidly converges to the fixed point 6174, known as the Kaprekar constant. Unlike other famous iterative procedures such as the Collatz function, the somewhat arbitrary nature of the Kaprekar routine doesn’t hint at fundamental mathematical discoveries yet to be made; rather, its charm lies in its intuitive definition (requiring no more than elementary mathematics,) its oddly off-center fixed point of 6174, and its surprisingly rapid convergence (which requires only five iterations on average and never more than seven.
In fact, it’s more like a weak anthropic principle - the only reason this particular combination caught Kaprekar’s attention is because it does happen to have a unique fixed point, while (for example) 3 or 5 digit numbers are ignored. However, there’s no real evidence of the Kaprekar routine converging to this mean or really even narrowing significantly across iterations, except by the inevitable winnowing as the number of unique values decreases as collisions occur. The observation that “diagonal neighbors” is the primary mechanism for collapses (other than digit sorting, obviously) is interesting, and it arose directly from doing the 100x100 grids and interpreting the patterns found there.
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