Example #1 Even, Odd, and Total Digits
  • Start with any number, say 9288759
  • Count the number of even digits,
    the number of odd digits,
    the total number of digits.
  • You get 347
  • Repeat and you get 123
  • Repeat and you get 123
  • Repeat and you get 123
  • Once you reach 123, you never get out, just as reaching a black hole in physics implies no escape.


Example #2 Words to Numerals (from Martin Gardner)
  • Start with any whole number, and write out its numeral in English.
    • Start with the number 5 -> FIVE
    • Count the number of characters in its spelling (count spaces and hyphens).
    • 4 -> FOUR
    • Repeat and you get 4.
    • Repeat and you get 4.
    • Once you reach 4, you never get out, so you have reached the black hole.
    • Start with the number 163 -> ONE-HUNDRED SIXTY-THREE
    • Count the number of characters in its spelling (count spaces and hyphens).
    • 23 -> 12 -> 6 -> 3 -> 5 -> 4
    • Repeat and you get 4.
    • Repeat and you get 4.
    • Once you reach 4, you never get out, so you have reached the black hole.


Example #3 Sum of the digits of the divisors
  • Start with any number greater than one,
    and write down all its divisors, including 1 and itself.
  • Now take the sum of the digits of these divisors (each individual digit!)
  • Eventually, you get 15, the black hole.
  • For example, start with 20.
  • The divisors are 1, 2, 4, 5, 10, and 20.
  • Adding the digits, you get 15.
  • The divisors are 1, 3, 5, and 15. The sum is 15, which repeats.