The 100th Letter

and

Other Counting Problems

INTRODUCTION

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

What is the 1998th digit in the decimal expansion of the fraction 1/7?

     Or here's another one that involves a more basic idea:

What is the 1000th 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 100th 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 100th 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 "100th" to "1000th" or "1,000,000th" 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.


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