Percentage of integers containing digit 3

What percentage of all integers contains at least one instance of the digit three? For example, 13, 31, 33 and 103 all contain the digit "three" at least once. ANSWER!   Answer: How Many Threes? 100% of all integers contain at least one three. What?!? How can this be? The solution is so surprising, it is difficult, if not impossible to believe that 100% of integers contain the digit three at least once.

The simple fact that the number 8, for example, has exactly zero threes in it seems to dispute this. Consider this: what percentage of the first ten numbers contains at least one three? That's easy- ten percent; three and only three. What percentage of the first one hundred number contains at least one  three? A slightly inflated nineteen percent. What percentage of the thousand numbers contains at least one three? Twenty-seven point one (27.1) percent.

The percentage of numbers with threes in them rises can be expressed as 1 – (.9)^n, where n is the  number of digits. It reaches 99% at about the point where n as 42 digits. The ratio of "threed" to "three-less" numbers at infinity would be 1 – (.9)^(Infinity), or 1. It is interesting to note that there are also an infinite number of integers which do not contain the digit  three. The simple progression "1, 11, 111, … " illustrates this fact.

This seeming paradox illustrates one of the many "problems" associated with trying to apply  concepts (like percentages) used for regular sets on the infinite. This puzzle, to the best of my knowledge, was originally posed by Clifford Pickover, the author and mathematician.

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