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### How Many Threes?

```
Date: 06/12/99 at 18:35:17
From: ken logan

Dear Dr. Math,

If all the numbers from 1 to 333,333 were written out, how many times
would the digit 3 be used?
```

```
Date: 06/13/99 at 08:07:50
From: Doctor Anthony
Subject: Re: Problem Solving (Graduate Level)

Consider the righthand column alone starting at 000000.

000000
000001
000002
......
333329
<-  there are 333330 rows to here
333330
333331
333332
333333

In the first column up to row 333330 there will be equal numbers of
all digits 0 to 9, so the number of 3's is 1/10th of this = 33333.
We then add the final 3 in this column, giving 33334 altogether.

Considering just the second column we have

000000
000001
......
333299
<-  333300 rows to here
333300
......
333310
......
333320
......
333330
333331
333332
333333

Considering the second column alone there are 1/10 * 333300 + 4 =
33334 threes, which is the same result as for the first column.
Similarly the other columns each give 33334 lots of 3's.

The total number of 3's is  6 x 33334 = 200004

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Discrete Mathematics

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