EK. The Law of Small Numbers

We answer last weeks puzzle, discuss the law of small numbers and ask again what is the smallest positive counting number that Google can’t find?

As a bonus we discuss the Fibonacci Substitution Sequence and close out with some Fibonacci Substitution Sequence Drumming by Akio Hizume.

To explore the sequence, we apply the following rules: replace each 0 with 001; replace each 1 with 01.

Beginning with 0, we next have 001. Next, applying our rules again, we have 001 001 01. In this way we obtain:

0
001
00100101
001001010010010100101

etc.

Amusingly the numbers of 0’s and 1’s at each step are a consecutive pair of fibonacci numbers, and the ratio of 0’s to 1’s, in the limit, is the famous golden ratio.

EDIT: We initially miswrote the substitution as
replace each 0 with 01; replace each 1 with 001, giving

0

01

01001

010010101001

01001010100101001010010101001

Stevestyle caught the error in the comments below

1. stevestyle said,

September 15, 2008 at 3:20 pm

Is the fibonacci thing true? I have the number of 0’s in the fourth line as being seven and it isn’t true from then on.

2. stevestyle said,

September 16, 2008 at 2:47 am

The number of 0’s, 1’s and the total number of digits all follow the rule: x(n+2) = 2x(n+1) + x(n).

The ratio of 0s to 1s is the square root of two.

3. strauss said,

September 17, 2008 at 1:13 pm

Another opportunity to catch an error in the Math Factor! Steve is on the ball and the ratio of 0’s to 1’s is indeed sqrt(2) to 1.

The fibonacci sequence appears in:

0->001
1->01

0
001
00100101
001001010010010100101
etc

4. strauss said,

September 18, 2008 at 7:53 am

Here are some of the nongoogleable numbers we received, suitably disguised:

J. Coleman ONE 8100110000 ZERO
M. Croucher ONE 92345211 NINE
D. Harris: ONE 177899871TWO
B. Tittle: ONE 22232512 ONE
T.I. Birkenes NINE 2323133 ONE
K. Siefkin: TWO 9036486 SEVEN

and the unmistakable winner in this round of searching:
M. Croucher: TWO 1753318 FOUR

There is an amusing elegance to some of these– Coleman’s entry for example, though large, surely is among the lowest nongoogleable numbers ending in a trail of five 0’s!

5. denken said,

October 4, 2008 at 4:51 am

There is an amusing elegance to some of these– Coleman’s entry for example, though large, surely is among the lowest nongoogleable numbers ending in a trail of five 0’s!

6. dfollett76 said,

November 8, 2008 at 9:45 am

The winner is no longer nongoogleable.

7. Casial said,

November 5, 2009 at 10:22 pm

In which movie the fibonacci sequence is part of the movie?
I cant remember the name *grrr*

RSS feed for comments on this post · TrackBack URL