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Ones or twos (Posted on 2011-12-07) Difficulty: 2 of 5
In a list of first 1,000,000,000 natural numbers each number is replaced by its digital root.
We get 1,000,000,000 one-digit numbers.

Are there more 1's or 2's ?

No Solution Yet Submitted by Ady TZIDON    
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Solution Depends what natural numbers are. | Comment 1 of 4
The digital roots just go in a cycle 1-2-3-4-5-6-7-8-9-1-2-3...  so for most values of n, the digital roots of the first n numbers contains equal numbers of 1's and 2's.  In fact because 1,000,000,000 divided by 9 equals 1, there will always be equal numbers of every digit except for one that will have an extra.

If we consider the natural numbers as beginning with 1, then the last number in the list is the digital root of 1,000,000,000 which is also 1, so there is one extra 1, so the answer is there are more 1's.

Some people seem to consider the natural numbers as beginning with 0 in which case the last number in the list is the digital root of 999,999,999 which is 9, so there is one extra 9 (no concern of ours), so the answer is there is the same number of both.

If someone, for some reason, considered the natural numbers as beginning with 2, they would conclude that there are more 2's

Edited on December 7, 2011, 1:48 pm
  Posted by Jer on 2011-12-07 11:02:43

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