No, that is not someone’s telephone number up there in the title of this piece. It is the name of a numerical puzzle guaranteed to spark wonder and amazement in the minds of your students. It is called Kaprekar’s (pronounced kuh-PREE-kur) Constant*. This little, mysterious math activity is one of a family of math procedures called Recurrent Operations.
Recurrent operations are repetitive procedures wherein each intermediate result (i.e. sum, difference, etc.) is used in turn until a specific outcome is ultimately obtained. The procedure to work the Kaprekar sort of “magic” is as follows:
1. Select a four-digit number, for example, 5634. 2. Form a new number from this number by rearranging the digits in decreasing order, e.g. 6543. 3. Reverse the number obtained in Step 2, then subtract the smaller number from the larger one. 6543 - 3456 3087 4. Take the difference just obtained and repeat the procedure in Step 2 and 3; that is, order the digits, reverse, and subtract. This should be continued until the surprise hits you. 8730 8532 7641 - 0378 - 2358 - 1467 8352 6174 6174
Do you notice that with 6174 the process comes to a “stand-still”? That is, 6174 just yields itself!
But the real surprise is yet to come.
Try the procedure on any other four-digit number. It always ends with 6174. This is a famous number in the field of recreational mathematics; it’s called Kaprekar’s Constant.
Students find this discovery quite fascinating and unexpected, so much so that they don’t even realize that one of your instructional objectives is to provide some practice in subtraction skills. But more importantly you can now use this opportunity as a springboard for further exploration by posing the following questions to them:
1. Who can find a number that requires the greatest number of subtractions to obtain 6174? 2. What happens if the Kaprekar process is applied to three- digit or five-digit numbers? (Dare we ask about still larger numbers?) 3. What happens if numbers in bases other than ten are used? For example, 4-digit numbers in "base 5". 4. Why does the game not work on such numbers as 3,333 or 888?
Many things could be said in favor of such activities from a pedagogical viewpoint, but only one will be mentioned here. Due to the strangeness of the whole situation and the interest created thereby, it causes the student to generate his own “problem set” of several exercises, based from only one or two starting numbers. That is to say, by asking him to do “only one or two problems”, he eventually does quite a few individual exercises — and enjoys it all the while.
One final note: if experience in systematic discovery is of more value to your instructional goals than merely subtraction practice, Questions #1 and #2 above make excellent investigations for pocket calculators.
*Discovered by Shri Dattathreya Ramachandra Kaprekar (1905-86?), a mathematician from India. He devised this activity in 1946 or 1949. See his photo on the right.
Additional websites that contain information about this topic:
- The Kaprekar Number and more playings with numbers
- Mudd Math Fun Facts: Kaprekar’s Constant
- Mathematical Black Holes
- Scientists of India – D.R. Kaprekar
- Sort, Reverse, Subtract
- Kaprekar Numbers
- More on Kaprekar’s Contants
- Mathews: Kaprekar Process (Highly recommended!)
Taken from T. Trotter, “Kaprekar”. Math Lab Matrix, Fall 1976, p. 8.