Unit 2 · Topic 05 · Number Play
Kabir picked any 4-digit number he liked. Anaya picked a completely different one.
Both followed the same simple steps. Both landed on the exact same number: 6174.
Every single time, no matter the starting number — as long as it wasn't all-one-digit.
"Pick any 4-digit number," said Ms. Rao, "as long as its digits aren't all identical, like 4444." Kabir picked 3524.
"Now arrange its digits to make the largest possible number, then the smallest possible number, and subtract the smaller from the larger."
Kabir arranged 3524's digits: largest is 5432, smallest is 2345. He subtracted: 5432 − 2345 = 3087.
"Repeat the exact same process on your new number." Kabir took 3087: largest arrangement is 8730, smallest is 0378 (378). 8730 − 378 = 8352.
"Again." 8352: largest 8532, smallest 2358. 8532 − 2358 = 6174.
"Once more." 6174: largest 7641, smallest 1467. 7641 − 1467 = 6174.
"It repeated itself!" said Kabir.
Meanwhile Anaya had picked a totally different starting number, 8091, and followed the identical steps. Largest−smallest, again and again. Within a few rounds, she also landed on 6174 — and then stayed there.
"That number, 6174, is called the Kaprekar constant, named after the mathematician D. R. Kaprekar who discovered it," said Ms. Rao. "Pick ANY 4-digit number whose digits are not all the same, and this exact process — largest arrangement minus smallest arrangement, repeated — always reaches 6174 within at most 7 rounds. Every single time."
Kabir tried to break it with 1000. Largest: 1000. Smallest: 0001 (1). 1000 − 1 = 999, which needs padding to 0999 for a 4-digit process. He kept going and, sure enough, still reached 6174.
"Why does it always land on the SAME number?" asked Anaya.
"That's the beautiful part," said Ms. Rao. "6174 is the one number where largest-arrangement-minus-smallest-arrangement gives back 6174 itself — a fixed point the process can't escape once it arrives. Mathematicians have proven this works for every valid 4-digit starting number, which makes 6174 one of the few numbers in mathematics with its own name and a fully proven, guaranteed property."
Kaprekar's routine: arrange a number's digits to form the largest possible number and the smallest possible number, then subtract the smaller from the larger.
Repeating this routine on ANY 4-digit number whose digits are not all identical always reaches 6174 within at most 7 rounds — and once reached, it stays at 6174 forever, since 6174 arranged-largest-minus-arranged-smallest gives 6174 again.
6174 is called the Kaprekar constant, named after mathematician D. R. Kaprekar, who discovered and proved this property.
Unlike the still-unsolved palindrome question, the Kaprekar constant is a fully PROVEN result — mathematicians have confirmed it works for every valid starting number, not just tested many examples.
Pick your own 4-digit number and apply Kaprekar's routine, counting how many rounds it takes to reach 6174.
Try a number with a repeating pattern like 1212 and see if it still reaches 6174.
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