I was reading a discussion on another message board concerning the series 1/1 - 1/2 + 1/3 - 1/4 + 1/5… Apparently, the sum of this series can be different values, depending on how the series is ordered. This lead to a couple different observations, one of which was this:

Suppose I order the natural numbers in a sequence like so:

1, 3, 2, 5, 7, 4, 9, 11, 6…(4i-3), (4i-1), 2i…

In this sequence, every natural number is acocunted for and has a particular place. Additionally, each even number is paired with two odd numbers, so obviously the ratio of even to odd numbers is 1:2. Equally obviously, I could construct a series in a different way to get any ratio of even:odd numbers that I desire. However, this conclusion bothers me mightily, since I’m used to even/odd numbers occuring alternately, with a 1:1 correspondence. So is the ratio of even:odd numbers undefined? Or am I overlooking something?

COROLLARY: Just previous to the even:odd observation, someone commented on the puzzle, How many threes? The puzzle is: What percentage of all integers contains at least one instance of the digit three? and the official answer is: 100%, the rationale being that the percentage of numbers with threes in them rises as the number of digits in a number rises. The percentage of numbers containing the digit three can be expressed as 1 - (.9)^n, where n is the number of digits. It reaches 99% at about the point where n has 42 digits. Since there are always more natural numbers greater than any given number than there are less than that number, it’s easy to see that, on average, all natural number contain the digit three.

However, this explanation relies on ordering numbers in the usual, counting way. If I construct a series like:

1, 3, 2, 13, 4, 23, 5, 30, 6, 31…[ith number not containing 3], [ith number containing 3]…

then, again, all natural numbers are accounted for, and it’s easy to see that only 50% of natural number contain the digit three.

Is the official answer of the “puzzle of the threes” still valid? Or not? For the same reason as the ratio of odd to even numbers is undefined?