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.*