# Trends in the distribution of reordered digits of Pi (OEIS A096566)

First let me try to describe in more details below the approach of "reordering" digits of Pi, which is used in OEIS A096566

https://oeis.org/A096566

and what I have done analyzing it so far.

I am looking at first 620 "reordered" digits (which is more than currently listed in A096566) of Pi (in decimal representation). The reordering is done in such way that, while looking at consecutive pipeline stream of digits, all "range" of 10 different decimal digits:

1,2,...,7,8,9,0

in their first occurrence are getting "collected", while all coming "repeating" digits of each kind (1,2,...,7,8,9,0 ) are getting "pushed back" to be written later ... .

Then the second "next" unique ten digits are getting collected, that is written, first looking for them in already "pushed back" group and then looking for coming-in (again with all "repeating" digits getting "pushed back") and so on until entire (second) set of all unique digits (1,2,...,7,8,9,0 ) is completely collected (written).

In total - I got 62 such sets (covering 62*10 decimal digits) - see all those 62 sets listed below - where each {....} line represents the set of such 10 digits collection.

 {3,1,4,5,9,2,6,8,7,0}
{1,5,3,9,2,8,4,6,7,0}
{5,9,3,2,6,4,8,1,7,0}
{3,2,9,5,8,4,1,6,7,0}
{3,2,8,9,5,1,7,4,6,0}
{3,9,5,8,2,4,7,1,6,0}
{3,9,4,5,2,8,6,0,1,7}
{3,9,4,2,8,6,5,1,0,7}
{3,9,2,8,4,6,1,0,5,7}
{9,3,8,2,4,6,1,0,5,7}
{9,8,3,2,4,6,0,5,1,7}
{9,8,3,2,6,4,0,5,1,7}
{9,8,2,3,4,6,0,5,1,7}
{9,8,2,3,4,5,0,1,6,7}
{8,2,9,3,4,1,5,0,6,7}
{8,9,2,3,4,1,0,5,6,7}
{8,2,3,9,4,0,1,5,6,7}
{8,2,4,9,3,1,0,5,6,7}
{8,2,1,5,9,4,3,0,6,7}
{8,2,4,9,5,3,1,0,6,7}
{8,2,4,9,1,5,3,6,0,7}
{8,2,4,9,5,3,1,6,0,7}
{8,2,9,4,5,3,1,6,0,7}
{2,8,4,9,3,5,1,6,0,7}
{8,2,9,4,3,1,5,6,0,7}
{8,4,2,9,1,5,3,6,0,7}
{8,4,2,9,3,6,1,5,0,7}
{8,4,2,9,3,6,1,5,0,7}
{8,2,4,6,3,9,1,0,5,7}
{8,4,2,3,6,9,1,5,0,7}
{8,4,2,3,6,1,5,9,0,7}
{8,4,2,3,6,1,9,5,0,7}
{8,4,2,6,3,1,9,5,0,7}
{4,8,2,6,9,1,3,5,0,7}
{4,2,8,6,1,3,9,0,5,7}
{4,2,8,6,9,3,1,0,5,7}
{4,8,2,6,1,3,0,5,9,7}
{4,8,2,6,3,0,1,5,9,7}
{4,8,2,3,6,1,5,0,9,7}
{8,2,4,6,3,1,0,5,9,7}
{2,4,8,6,1,3,0,5,9,7}
{2,4,8,1,6,3,5,9,0,7}
{8,2,4,1,3,6,5,9,0,7}
{2,8,4,1,5,3,6,9,0,7}
{4,2,1,8,3,6,9,5,0,7}
{4,2,1,8,3,5,6,9,0,7}
{4,1,2,3,8,9,6,5,0,7}
{1,4,8,2,3,9,6,5,0,7}
{1,4,2,3,8,5,9,6,0,7}
{1,4,8,5,2,9,3,6,0,7}
{1,4,2,8,3,9,6,5,0,7}
{1,4,2,8,9,3,6,5,0,7}
{1,2,8,4,9,3,6,0,5,7}
{1,2,8,3,4,6,9,5,0,7}
{1,3,2,4,8,9,6,5,0,7}
{1,4,3,2,9,8,6,0,5,7}
{1,3,4,2,9,6,8,0,5,7}
{1,4,3,2,9,6,8,5,0,7}
{1,4,3,2,9,6,8,5,0,7}
{1,4,3,2,9,6,8,5,7,0}
{1,3,2,9,4,6,8,7,0,5}
{1,2,3,4,6,9,8,7,0,5}


Here are results of arithmetic averages which I got for each (out of ten) positions between those 62 sets:

first position digits average 4.694538

second position digits average 4.306452

third position digits average 4

forth position digits average 5.048387097

fifth position digits average 4.951612903

sixth position digits average 4.548387097

seventh position digits average 4.161290323

eights position digits average 4.274193548

ninths position digits average 2.870967742

tenth position digits average 6.14516129

Above results show that for the first eight positions, digits in each position were changing from the set to set progressively more and more randomly - thus averages for those positions are getting closer to 4.5 .

However for positions 9 and 10 - such randomization was not achieved yet within first 62 sets ...(though looking outside of presented so far 62 sets data and relying on the known observation that eventually the average digit value in the decimal expansion of Pi comes practically to 4.5), I could "speculate" in advance that it will come to 4.5 eventually and for positions 9 and 10 too ... - but it looks like that positions 9 and 10 are "randomizing" at much slowly rate than the other 8 positions ... and that might be (or might not be) interesting.

I am not sure how many more sets (beyond 62, which I presented here) are needed to get arithmetic averages for "ninth" and "tenth" digit positions (within the set) to reach the same proximity of 4.5 for average value, as it is achieved already by first 8 positions within those 62 sets ... .

It is also notable that if to average positions 9 and 10 together, the average between those two, within available so far 62 sets, will be close to 4.5.

# Conclusion and questions

There appears to be that those first 62 sets listed above have a slight hint of retaining some loose organizational order between predecessor sets and successor sets.

But I presume that further "down the road", beyond the first 62 sets, one will see that gradually the level of randomness in sets digits composition order is increasing and adjacent sets become more more disconnected from each other.

What I am trying to say that in case of digits Pi (after applied above discussed reordering) it appears that there exists some sort of transition from initial order (within first 620 digits) to total randomness ...

I used Maximal information-based nonparametric exploration statistical analysis program (MINE) by "David Reshef at al ".

Being applied (by me ) "pairwise" to the first 62 terms, MINE shows high values (up to 1) of the maximal information coefficient (MIC), which is a measure of two-variable dependence designed specifically for rapid exploration of many-dimensional data sets.

The links (thanks to LVK for the upload) to the excel spreadsheet, turned into the comma separated value file (.csv), with the data (62 sets of reordered Pi digits) and the MINE generated output .csv file, which was produced (at my home PC Windows based computer) upon executing

java -jar MINE.jar PiReordered.csv -allPairs cv=1.0

correspondingly are

https://dl.dropbox.com/u/29863189/PiReordered.csv

and

https://dl.dropbox.com/u/29863189/PiReordered.csv%2Callpairs%2Ccv%3D1.0%2CB%3Dn%5E0.6%2CResults.csv

Does such concept of transition from order to randomness exist ?

Could this above observation be statistically confirmed or disproved ?

If "yes" - what specific tools / methods of statistical analysis could be applied ?

I also received suggestion that in order to test whether this discussed above feature is only characteristic to (some initial digits of) Pi, the same reordering should be applied to the some significant number of randomly generated very long strings of decimal digits - to see if there the same pattern behavior will appear or not - is it useful ?

Thanks,

Best Regards,

Alexander R. Povolotsky

PS - in response to LVK's answer and his comment, which I am quotting here "The nth line of your table consists of the digits written in the order of their nth appearance in π. This could be in principle read off the graph by crossing it with the horizontal line y=n and reading off the intersection points from left to right. (In practice this is not convenient due to the low resolution and overlap between the curves.) ..... I don't think there is any statistical method for analysis of the data organized in this way. You'll probably need to devise one yourself. ..... – LVK Sep 10 at 16:02"

LVK - thanks for your thoughts and valuable contribution ! I think though that the frequency chart somewhat hides away the positional dependency between the unique digits of the {1,2,...,9,0} set. The table presentation with the columns representing the particular combination of (all) digits (from 1 to 0 in above mentioned set) for each consecutive ten digit collection is, in my opinion, more revealing in that regard.

My questions still remain to be in place:

1) is there some "other" (I would call it "transitional") non-randomness exists for some few hundreds "initial" digits of Pi (beyond the order imposed by the re-arrangement itself), which is getting revealed by this re-arrangement ?

2) what (other than MINE) quantative statistical methods/tools could be used in analysis of this situation -

PPS I am trying to rework first 3 columns in already posted MINE results csv file (where first two columns are "textually enumerated" names of the 10 digitt sets, like for example "18thSet", and the 3rd column is MIC value for the two sets identified in the first two columns at the same row) into three (3)-dimensional "surface" chart with each column, mentioned above, be correspondingly x,y and z values ...

Doing it manually via converting into table -- by keeping the first column in tact, transposing the second column into up-most row and filling the table's body by the MIC values from the 3rd column is very laborious.

I found discussion at

https://stackoverflow.com/questions/7083044/mathematica-csv-to-multidimensional-charts

how to do it with Mathematica, but I don't have it ...

Could some one (who has Mathematica) be kind enough to do it (and post) ?

• There will be high correlation between positions in the $n$th set and the $n+1$th set, and this correlation will increase with $n$. So you should not expect a "transition from order to randomness" Apr 11, 2014 at 10:29
• As Henry says, there will be a strong correlation between the $n$th set and the $n+1$th set, but it will go away after *many* sets. Better than calculating the average at each position it would be better to find the greatest and least number at each position and how many occurrences they each have. You can download $\pi$ to millions or billions of places to improve the statistics. Jan 31, 2023 at 4:17

I also received suggestion that in order to test whether this discussed above feature is only characteristic to (some initial digits of) Pi, the same reordering should be applied to the some significant number of randomly generated very long strings of decimal digits - to see if there the same pattern behavior will appear or not - is it useful ?

Yes, this is an easy numerical experiment that could help you understand just how likely such patterns are to emerge under your process of rearrangement.

In private communication I received the following: "It is rather obviously non-random. For example, the digit 7 appears at the end of 53/62 sets...

Of course it's non-random. To begin with, every digit appears exactly once in every line. This already makes this set non-random, but we know that this non-randomness is the result of rearrangement. As for the digit 7, it simply lags in frequency in the beginning of pi, which is what places it at the end of your 10-digit groups. Here is the chart of digit counts in $\pi$ which I just made in a spreadsheet. Also a link to full size version and to the actual OpenOffice spreadsheet that I used.

• If you think it's silly to do this kind of thing in a spreadsheet, you may be right... I just wanted to do it quickly. Feel free to contribute Python code or/and a link to a similar chart that is already available online (I could not find one).
– user31373
Sep 10, 2012 at 3:01
• A tool that lets you quickly and easily do exactly what you want is almost always the right tool for the job.
– user14972
Sep 14, 2012 at 23:56

"some sort of transition from initial order (within first 620 digits) to total randomness ... Does such concept of transition from order to randomness exist ?"

If the phenomenon is that pi digit summaries are different from your expectation when you use a small sample size but are close to your expectation when you use a large sample size, then maybe various laws of large numbers could explain it. The variance of the sample mean is smaller when the sample size increases.

• In private communication I received the following: "It is rather obviously non-random. For example, the digit 7appears at the end of 53/62 sets, which is very, very, very unlikely if the location of 7 or its frequency in the set were random. There is also a high degree in autocorrelation in other positions. So as far as sequential data is concerned it is totally not random, nor is it random by column" .
– Alex
Sep 9, 2012 at 22:11
• Above approach (suggested in the BINN's comment) assumes, that the given sequence of digits is random. However, the randomness of digits of Pi is not proved - to the best of my knowledge and, in this particular case, some order is imposed by described rearrangement. In such situations, I am afraid that "strict statistical approach towards increasing of the "sample" size could lead to "throwing the baby together with the basing water"
– Alex
Sep 16, 2012 at 15:57