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For ? Day. 2015, Wolfram Research stirred up publicity with the blog "Pi or Pie?! Celebrating Pi Day of the Century (And How to Get Your Very Own Piece of Pi)":
https://blog.stephenwolfram.com/2015/03/pi-or-pie-celebrating-pi-day-of-the-centuryand-how-to-get-your-very-own-piece-of-pi
I grumbled that no self-respecting Deity would bother sending clues to worshipers dumb enough to use decimal instead of continued fractions (CF). However, in 2019 we're barely able to afford a full CF version of Wolfram's birthday games. Assuming GaussKuzmin distribution, define
![enter image description here](https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2019-01-02at16.04.32.png&userId=20103)
Then
![enter image description here](https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2019-01-02at16.16.23.png&userId=20103)
I.e, 41.5% of terms should be 1, 17% should be 2, etc. But cfprob also gives us the probabilities of term sequences:
![enter image description here](https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2019-01-03at10.55.36.png&userId=20103)
(Invariant under reversal but not shuffling.)
This says to expect about six 1,2,3's in every burst of 1000 terms:
Try a million :
![enter image description here](https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2019-01-02at16.13.08.png&userId=20103)
(0.5 seconds for a million terms. Have I actually lived to see this?)
![enter image description here](https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2019-01-02at16.17.57.png&userId=20103)
Continued fractions accommodate fancier date formats:
![enter image description here](https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2019-01-02at16.18.54.png&userId=20103)
Sure enough, there were two of them.
But for really fancy dates,
![enter image description here](https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2019-01-02at16.20.56.png&userId=20103)
we'll need Eric Weisstein's record ? CF calculation.
Utility functions
These are the definitions of the utility function tim you will need for the above evaluations:
![enter image description here](https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2019-01-02at16.21.33.png&userId=20103)
Attachments: