This article builds on ideas written about in three earlier blog posting: Pi, Passwords (specifically PIN numbers), and De Bruijn sequences. Ingredient 1: In 2013 I wrote an article about how “spectacularly unimaginative” people are when selecting their passwords. I even got asked to turn this into a TED |

Photo Credit: Jessica Snow

Ingredient 2: A little earlier, I’d preformed some whimsical analysis on the digits of Pi. What I was trying to show was that, if you look hard enough (and long enough), you can find anything you want in the digits of Pi (probably). Your age, your phone number, your street address, your social security number … … even your ATM card PIN! |

Ingredient 3: De Bruijn sequences. Using the same principles I demonstrating with De Bruijn sequences (using a sliding window of contiguous digits), I was curious how many digits of Pi would you need so that every four digit PIN would be encountered at least once. |

If we contrived an automated machine with a robot arm, and fed it Pi, how many key presses would it need (worst case) to go through all possible four digit PINs? (And, out of curiosity, What would be that last PIN? Which of the 10,000 four digit permutations occurs last in Pi?) The first four digit PIN is |

The next is Each time we move the slider along we encounter a PIN that comprises three digits from the last PIN (shifted left), plus a new digit on the end. Obviously, there will be some repetition. In fact the first repetition occurs very early on. The digits |

Here is a scatter plot of the results. On the *x-axis* we have each of the PINS (from **0000** on the left to **9999** on the right). The *y-axis* shows at what position in the decimal digits this PIN first occurs at.

Interestingly, there is one *massive* outlayer, and this is the PIN **6716**. This occurs first starting at digit position *#99,846*.

So, to be sure of getting every four digit password, then Pi to at least 99,847 digits is needed (strictly speaking 99,848 digits, as the digit following the last one is a six, and without an extra digit of precision it would get round up!)

(The shortest possible arrangement of digits is 10,003 characters).

If we expand the analysis to five digit PINs, then the furthest distance discrete PIN is **33394** which occurs way out at *#1,369,560*.

For six digit PINs, the furthest distance discrete PIN is **569540** which occurs way out at *#14,118,307*.

If you are looking for when **8675309** occurs, than I'm pleased to tell you it first appears at position *#9,202,591* of Pi.

You can find a complete list of all the articles here.^{} Click here to receive email alerts on new articles.