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## Usenet Newsgroups : rec.puzzles Hall of Fame

The rec.puzzles Hall of Fame is a compilation of over 500 of the most popular puzzles that have been posted and discussed in the rec.puzzles newsgroup. In most cases a detailed solution has been provided.

Many of these puzzles also appear in Braingle's own collection.

 rec.puzzles Newsgroup FAQ Hall of Fame alt.brain.teasers Newsgroup FAQ
Categories : pickover : pickover.03.p

```Title: Cliff Puzzle 3: Too many 3's
From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please include your name,
address, affiliation, e-mail address.  If you like, tell me a little bit
about yourself.  You might also directly mail me a copy of your response
in addition to any responding you do in the newsgroup.  I will assume it
is OK to describe your answer in any article or publication I may write
in the future, with attribution to you, unless you state otherwise.
Thanks, Cliff Pickover

* * *

How many numbers have at least one digit -- a three?

In the first 10 numbers, 1,2,3,4,5,6,7,8,9,10 there is only one number
which contain the digit 3.  This means that 1/10 or 10% of the numbers
have the number 1 in the first 10 numbers.  In the first 100 numbers the
occurrence of numbers with at least one three seems to be growing.  In
fact there are 19 numbers:  3,13,23,33,43,53,63,73,83,93,
30,31,32,34,35,36,37,38,39.  This means that about 19% of the digits
contain the number 3 in the first 100 numbers.

We can make a table showing the percentage of numbers with
at least one 3-digit for the first N numbers.
N        %
10       1
100      19
1000     27
10000    34

The percentages rapidly increase to 100% indicating that almost all of
the numbers have a 3 in them!  In fact, a formula describing the
proportion of 3's can be written:  1-(9/10)**N.  The proportion gets
very close to 1 as N increases.

Stop And Think

1. How can it be that almost all of the numbers have a 3 in them?

```

Solution

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