# Abstract Confusions

Complexity is not a cause of confusion. It is a result of it.

2010 is gone. And I always like even numbered years compared to odd ones. I was asking to myself, what’s special about 2011? It works out that 2011 is indeed special.

First: 2011 is a prime number. The fundamental building blocks of number system, prime numbers are special. It means 2011 can not be expressed a product of smaller prime numbers (or any other numbers for that sake). And then, a friend pointed out 2011 can be expressed as sum of 11 prime numbers.

Further, the number 2011 can also be represented as sum of three consecutive primes as well (661+673+677=2011).

And then, it is also notable, 2011 is a toothpick number.

### Toothpick Number

Toothpick numbers, or toothpick sequences are the number of toothpicks used in particular toothpick arrangement. The steps for generating toothpick sequence is

• Start with a single two dimensional toothpick. Placed on a plane. Parallel to ‘Y’ axis.
• Place two more toothpicks, one each at the end of a toothpick.
• Keep repeating, until you maintain a fractal like symmetric shape.
• the number of toothpicks used in any step is the toothpick number of that state and the sequence is toothpick sequence.
The steps would look like below.

toothpick number 1 to 10

The structure would look like below after 56th step.

Toothpick number - t(56)

It is difficult to find out how many toothpicks are needed for $n^{th}$ step. And there are generating functions on this. Read this paper for a serious treatment.

Often the toothpick setup is studied in reference with Cellular Automata, Conway’s game of life and universal Turing machine.

Try out several variations online (also called as toothpick movie version).

Coming back, 2011 is the number of toothpicks at step 60, it is 25th prime number in toothpick sequence.

Isn’t that cool? Happy new year to all!

### 8 Responses to What’s special about 2011?

1. Omar January 8, 2011 at 2:07 AM

2011 is the number of toothpicks after 60 stages.

• SREE GURUPARAN January 8, 2011 at 10:13 AM

@Omar, check here: http://oeis.org/A139253/b139253.txt , it shows the toothpick sequences for n starting from 1 through 181. The step mentioned in the post is +1, considering the fact that whether you start with 1 at step 0 or step 1.

+—————+
| n | a(n) |
+—————+
| 25 | 2011 |
+—————+

• SREE GURUPARAN January 8, 2011 at 10:19 AM

@Omar, You are right, I referred to “primes in the toothpick sequence”. Obviously there are missing numbers in it.. Thanks for the spotting. Provide a link if you have (I will link). Cheers!

2. Golmes February 1, 2011 at 6:58 AM

we(what we? should be you!) talk about this next time we meet da – btw because of the contents u’ve put, you should be buying

3. Raja February 11, 2011 at 11:42 AM