Mathemagical Black Holes
by Dr. Mike Ecker in Spring 2004 Recreational & Educational Computing

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.

Send any comments or questions to: David Pleacher