Pi, PINs and Permutations | Hacker News Reader

Pi, PINs and Permutations

datagenetics.com

25 pointssqueakynick14 comments

If you want to play more with it and try to found when all 7 digit combination of pi repeat themself, I build a Pi as a service API for pi day. It can serve the first 1 billions digit of pi over http.

http://piaas.org/

I love Datagenetics posts. Related:

https://github.com/philipl/pifs

6716 confirmed most secure 4-digit PIN.

aha! not anymore now that we know ;-D

Why is the first pin 1415 and not 3141?

[edit to add] and I don't mean that in a snarky way. Was just curious.

No reason really. I just arbitrarily started using the digits after the decimal point (to avoid the confusion of what to do with the ".")

Other Pi sites seem to follow similar protocol. It will create an "off-by-one" for most answers, but since the string that is the furthest away does not start with a "3", other than the off-by-one, it will not change the results :)

Just a random question that popped into my head while reading (thank you for the article!)... What would be the longest repetitions of digits within various orders of magnitude of Pi?

In other words, the first 10 characters (again looking at only the fractional portion:

1415926535

The longest repetition is a length of one and the digit is 1.

Playing with regexes, I found:

In 100 digits, you get a repetition of the digits 592

In 1000 digits, the first longest repetition is 23846

In 10000 digits, I found a six digit repetition, 120190 but at that point, the quadradic nature of the regex made searching for 7 impossible so I would probably write a program that used something like suffix trees to get further.

There is this service to search in the digits: http://www.angio.net/pi/piquery

Is there any statistical/probabilistic explanation of why the answer is almost 100'000 ?

Probably, since five digits requires 1,369,560 decimals and six digits 14,118,307.

Each extra digit seems to require ~ ten times as many decimals.

If you approximate the positions of the pin numbers as independent exponential distributions with parameter λ = 1/10000. Then the expected value of the maximum is the sum of 10000/k for k = 1 to 10000 [1], which is approximately 1000*(log(1000)+γ) with γ = 0.57721... [2]. This gives the following values for the expected number of digits needed to get all k-digit pin numbers:

k | n

------------

4 | 97,875.6

5 | 1,209,014.1

6 | 14,392,726.2

7 | 166,953,113.2

8 | 1,899,789,640.9

Which agrees reasonably well with the actual values, at least those found so far.

[1] http://math.stackexchange.com/questions/80475/kth-order-stat... [2] https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_const...

There's no proof that pi contains every possible number combinations. It's a common misconception that somewhere in pi you'll find everything.

One of the other articles referenced, about De Bruijn sequences, is much more interesting http://www.datagenetics.com/blog/october22013/index.html

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