Numbers of No Escape

Start with any natural number, such as 69534891. Count the number of even digits, the number of odd digits, and the total number of digits. In this case, there are three evens, five odds, and a total of eight digits. Use these three numbers as digits to form a new number: 358.

Repeat the steps with the new number, counting evens, odds, and the total number of digits. You get 123. If you perform the same set of operations on 123, you get 123 again.

Try another number: 141592653589793238462643383279502884197169399375105820974. Counting 0 as even, there are 24 evens, 33 odds, and 57 digits in total. Applying the process to 243357 gives 246, then 303, then 123.

In fact, no matter what number you start with, this iterative process always leads to 123.

Michael W. Ecker describes the number 123 as a “mathemagical black hole” with respect to this particular process. “Once you hit 123, you never get out,” he says, “just as reaching a black hole of physics implies no escape.”

Ecker is a mathematician at Pennsylvania State University, Wilkes-Barre Campus and publisher of the lively newsletter Recreational & Educational Computing. He formulated the notion of a mathemagical black hole in the mid-1980s and has been looking for examples of such iterative surprises ever since.

“A mathemagical black hole is, loosely, any element to which other elements are drawn by some stated process,” Ecker says. “Though the number itself is the star of the show, the real trick is in finding interesting processes.”

Why do such mathemagical black holes occur? In general, processes that turn large inputs into significantly smaller outputs can quickly reduce even an infinite universe of starting points to a manageable, finite set of cases.

For the 123 black hole, for example, any given number eventually reduces to a three-digit string, and this three-digit string must go to 033, 123, 213, or 303. Each of these possibilities ends up at 123.

Here’s another curious example, credited to Martin Gardner. Take any whole number and write out its name in English. For example, 5 is FIVE. Count the number of characters in the spelled-out name: 4. Spell out 4: FOUR. Repeating the process gives 4, again . . . and again and again.

Now try, say, 163. Spell out the number: ONE HUNDRED SIXTY-THREE. Including hyphens and spaces, the name has 23 characters. TWENTY-THREE, in turn, gives 12, then 6, 3, 5, and finally 4.

In this case, 4 is the mathemagical black hole. Numbers in other languages have the same property, though not necessarily with 4 as the black hole.

Here’s another example, courtesy of Mike Ecker. The number 153 equals the sum of the cubes of its digits (1³ + 5³ + 3³ = 153). Here’s an iterative process that produces 153 as the black hole.

Start with any positive whole number that’s a multiple of 3. One at a time, calculate the cube of each digit. Add up the cubes to form a new number. Then repeat the process. You eventually reach 153.

For instance, starting with 432 (a multiple of 3), you get 4³ + 3³ + 2³, or 99. The new number, 99, leads to 1458, then 702, 351, and 153. Note that each successive number is divisible by 3.

There are many more examples. Ecker has built up an extensive and varied collection of recipes that lead to mathemagical black holes in all sorts of settings. And he’s always looking for more!