A Numerical Peculiarity
Created | Updated Apr 28, 2004
What follows is one of the more peculiar features of numbers in general. Think of a string of digits. Write them down. Underneath, write the total number of even digits, the total number of odd digits, and the total number of digits. Do the same procedure to this number. You should now have the number 123; if you do not, keep going until you do. Zero should be counted as an even number, though in many cases, it also works as an odd one. This procedure done to any number will give 123.
A FEW EXAMPLES
27825 ---> 325 ---> 123
238751639827 ---> 5712 ---> 224 ---> 033 ---> 123
7539175 ---> 077 ---> 123
248286 ---> 606 ---> 213 ---> 123
200779 ---> 336 ---> 123
42 ---> 202 ---> 033 ---> 213 ---> 123
123 ---> 123
Here is the proof:
1) It works for all single digits, as there is then either
1a) The digit is even, so one even digit, no odd digits, and one digit in total, then 101--->123, or
1b) The digit is odd, so no even digits, one odd digit, and one digit in total, then 011--->123
2) It works for all two digit numbers, as there are either
2a) Two even digits, then 202--->303--->123, or
2b) Two odd digits, then 022--->303--->123, or
2c) One even digit and one odd digit, then 112->123
3) It works for all three digit numbers, as there are either
3a) Three even digits, no odd digits, then 303--->123, or
3b) Two even digits, one odd digit, then 213--->123, or
3c) One even digit, two odd digits, then 123, or
3d) No even digits, three odd digits, then 033--->123
4) It works for all numbers with nine digits or less, as the number of even digits is less than or equal to nine, the number of odd digits is less then or equal to nine, and the number of digits is less than or equal to nine. Thus the first number produced by the rule does not exceed 999, and this results in 123 by the above.
5) It works for all numbers with less than one thousand digits, as the even, odd, total digit counts each have at most 3 digits, which when concatenated give a number with 9 digits or less, which works by the above.
6) The principle of induction can be used, to make further rules in the same way the rule 5 extended rule 4, so is true for any number of digits.
A FEW EXAMPLES
27825 ---> 325 ---> 123
238751639827 ---> 5712 ---> 224 ---> 033 ---> 123
7539175 ---> 077 ---> 123
248286 ---> 606 ---> 213 ---> 123
200779 ---> 336 ---> 123
42 ---> 202 ---> 033 ---> 213 ---> 123
123 ---> 123
Here is the proof:
1) It works for all single digits, as there is then either
1a) The digit is even, so one even digit, no odd digits, and one digit in total, then 101--->123, or
1b) The digit is odd, so no even digits, one odd digit, and one digit in total, then 011--->123
2) It works for all two digit numbers, as there are either
2a) Two even digits, then 202--->303--->123, or
2b) Two odd digits, then 022--->303--->123, or
2c) One even digit and one odd digit, then 112->123
3) It works for all three digit numbers, as there are either
3a) Three even digits, no odd digits, then 303--->123, or
3b) Two even digits, one odd digit, then 213--->123, or
3c) One even digit, two odd digits, then 123, or
3d) No even digits, three odd digits, then 033--->123
4) It works for all numbers with nine digits or less, as the number of even digits is less than or equal to nine, the number of odd digits is less then or equal to nine, and the number of digits is less than or equal to nine. Thus the first number produced by the rule does not exceed 999, and this results in 123 by the above.
5) It works for all numbers with less than one thousand digits, as the even, odd, total digit counts each have at most 3 digits, which when concatenated give a number with 9 digits or less, which works by the above.
6) The principle of induction can be used, to make further rules in the same way the rule 5 extended rule 4, so is true for any number of digits.
