Monday Morning Math: D. R. Kaprekar and his constant

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Happy May Day!  It’s finals week here, and so this will be the last Monday Morning Math until sometime in September. But in the meantime, here’s a neat magic trick

• Start with a 4 digit number. The digits don’t have to all be different, but they can’t all be the same (so 1001 is OK but 1111 is not).
• Now look at the 4 digits, and write down both the largest number and the smallest number you can make with those digits: in my example the largest would be 1100 and the smallest 0011.
• Take the difference:  I got 1100-0011, which is 1089.  (Hey, that’s pretty cool!  I’ve seen that number before.  But I digress.)
• Now repeat the process: take your new 4 digit number, write the largest and the smallest number you can make with those digits and take the difference.  And keep doing it.  After a while, you’ll get the same number over and over again.  And that number is……

6174

This is called Kaprekar’s constant, named after Dattatreya Ramchandra Kaprekar.  D. R. Kaprekar was born on January 17, 1905, in Dahanu, India, and enjoyed math puzzles and numbers from when he was young.  He studied math in college and in 1927, when he was 22, was awarded the R. P. Paranjpe Mathematical Prize for original work.  His discovery of the pattern above isn’t his only work, but it might be the best suited for magic tricks.

There’s a  version with 3 digits too,  but once you get to 5 or more digits you start getting cycles instead of the same number over and over.  And that’s just Base Ten – you can explore other bases to see what happens. Enjoy!

Sources:

• Wikipedia
• MacTutor (where the picture came from)
• Hat tip to Ricardo Teixeira, whose presentation on mathemagic introduced me to this constant!