**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.

order generic viagra canada order viagra canada online viagra without perscription non prescription viagra canada generic viagra canada viagra non prescription order viagra canada generic viagra usa order generic viagra online non prescription viagra viagra non prescription non prescription viagra usa
online viagra without perscription non prescription viagra online viagra non prescription viagra non prescription order viagra usa order generic viagra canada viagra without perscription canada non prescription viagra online order generic viagra online order generic viagra canada viagra without perscription canada order generic viagra canada