**The 100**^{th} Letter

**and**

**Other Counting Problems**

**INTRODUCTION**

A popular category of puzzle-problems often involves something like the following:

**What is the 1998**^{th} digit in the decimal expansion of the fraction 1/7?
Or here's another one that involves a more basic idea:

**What is the 1000**^{th} digit of this sequence:
**
1234567891011121314151617... ?**

In another context, I have seen a problem like this one that uses letters instead of numbers:

**What is the 100**^{th} letter in this sequence:
**
ABBCCCDDDD... ?**

**MY PROBLEMS**

I began thinking about this idea the other day and came up with a problem that I had never seen before in any book, magazine, or contest exam. It goes like this:

*If the word names of the counting numbers are written out as "one, two, three, ... ", what letter in which number name would be the 100*^{th} letter?

Of course, a natural companion of that problem would be this:

*If the word names of the first 100 counting numbers were written out as "one, two, three,..., one hundred", how many letters would be written?*

Now, dear reader, don't be deceived by the apparent simplicity of these two questions. To paraphrase a popular saying: *ease of solution lies in the mind of the solver*. These problems were given to my 7th grade students recently [1998]. There was plenty of challenge here for these individuals, especially for the second problem. The results were interesting; some were even correct. Many different strategies were employed. And the important thing was that __all__ would have been successful, except for a little error "here and there".

And if you want a greater challenge, just increase the number "100^{th}" to "1000^{th}" or "1,000,000^{th}" in the first problem; and likewise "one hundred" to "one thousand" or "one million" in the second problem.

I leave it to you to solve these letter problems.

E-mail me with
your solutions and explanation of how you arrived at your answers.