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.
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Example #2 Words to Numerals (from Martin Gardner)
- Start with any whole number, and write out its numeral in English.
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- 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.
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- 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.