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 AUGUST 2005

Getting into an Endless Loop
By Alfred Posamentier, Ph.D.

Now that the summer is upon us, it is time for some true recreation—in mathematics, of course. In that spirit you may want to consider a rather unique situation that demonstrates an unusual phenomenon that arises out of the peculiarities of our decimal number system. There isn’t much you can do with it, other than to marvel at the outcome. This amazing relationship is not something we can prove true for all cases, yet no numbers have been found for which it won’t work. That, in itself, suffices to establish that it is apparently always true. You may wish to have your students use a calculator, unless you want them to practice subtraction. It is best to go through this short procedure by yourself, to really get an appreciation for it. Here is how this procedure goes:

Begin by having them select a four-digit number (except one that has all digits the same).

Rearrange the digits of the number so that they form the largest number possible.

Then rearrange the digits of the number so that they form the smallest number possible.

Subtract these two numbers (obviously, the smaller from the larger).

Take this difference and continue the process, over and over and over, until you notice something disturbing happening. (Don’t give up before something unusual happens.)

Eventually you will arrive at the number 6,174, perhaps after one subtraction, or after several subtractions. When you do, you will find yourself in an endless loop. Rest assured eventually you will reach this number (don’t despair!).

When you do this with your students and they reached the loop, remind them that they began with a randomly selected number. Isn’t this quite an astonishing result? Some students might be motivated to investigate this further. Others will just sit back in awe. Either way they have been charmed again with the beauty of mathematics.

Here is an example of this activity: We will (randomly) select the number 3,203.
The largest number formed with these digits is: 3320
The smallest number formed with these digits is: 0233
The difference is: 3087
The largest number formed with these digits is: 8730
The smallest number formed with these digits is: 0378
The difference is: 8352
The largest number formed with these digits is: 8532
The smallest number formed with these digits is: 2358
The difference is: 6174
The largest number formed with these digits is: 7641
The smallest number formed with these digits is: 1467
The difference is: 6174
And so the loop is formed, since you keep on getting 6174 if you continue.
Enjoy the summer with a good book on recreational mathematics that will further demonstrate the beauty of mathematics. Further recreational reading: Math Charmers: Tantilizing Tidbit for the Mind (Prometheus, 2003), and Pi: A Biography of the World’s Most Myterious Number (Prometheus, 2004)#

Dr. Alfred S. Posamentier is Dean of the School of Education at City College of NY, author of over 35 books on math, and member of the NYS Standards Committee on Math.