==> 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?