TDLR
- From 1 to 9, all Mancala Identity Boards (IB) converge.
- From 1 to 1000, more IB loop (55.8%) versus converge (44.2%).
- The pattern seems mostly random, but from a Chi-Squared analysis of binary sequences, looping IB are more likely to happen in clusters, while converging IB are more likely to happen standalone.
- A relatively high frequency of looping IB happens at Mod(X, 10) = 2.
Dashboard:
https://movingscreen.net/identity-board-looping-and-converging/
Mobile Dashboard:
https://movingscreen.net/identity-board-looping-and-converging-mobile/
AppropriatelyLDidR:
I’ve written about my homeopathic Mancala mathematics before. Not in this website, necessarily, but rather on infinity-scroll Microsoft Word document I maintain as an overrun mangrove of my divers contemplations.
A la Creed in The Office: to protect the world from the horrors of my headspace, of course.
In any case, even though I can’t provide any links to adduce my assertion, Mancala algorithms are simple enough. You’ll get the gist within a couple of seconds.
For the flavor du jour of rabbit hole rumination, I’m interested in what happens to the looping / converging of Identity Boards. What is an Identity Board, you have no conscious interest in asking, and yet probably did in a manner akin to when the doctor takes that mini-mallet and smashes it against your knee for no good reason?
A Mancala configuration of size n where all elements are 1.
So:
[1, 1] or
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] or
[1] or
[1, 1, 1, 1, 1]
All Identity Boards (IB).
From that we ask, which IB converge? Which loop? Does any pattern emerge as we make IB bigger and bigger?
First, IB of size 2 (2-IB), 3-IB, and 4-IB to break the tension:
They all converge! What joy, diletto straodinario.
I’ll spare you the banality of unnecessary spreadsheet art. I’ll skip IB 4, 5, 6, and 7, on collateral of Scout’s Honor you can believe that all the beans end up in one bucket. (I was never a Boy Scout, neither here nor there). 8-IB, coming up:
That settles it, a generic IB of any size always converges to a singularity. Start polishing that Fields Medal now.
But hold on a minute! We are scientists of the utmost scruples here. We are bound to trade in our hard-earned Dogecoin for a larger sample size, in the interest of statistical reliability. So we’ll do a little more research.
10-IB. Surely this will be sufficient, the War to End All Wars.
Angry face. My quarrel was not with Mancala, but with God.
The loop from 3 to 7, 7 to 3, these numbers looming like twin pillars in the Judeo-Christian imagination.
So at the turn of the order of magnitude, things fall apart. Now we know.
The pretense of all streams of life coalescing into one beautiful magnificent ocean of consciousness has been shattered, but nevertheless we solider on. If for no other reason than it is somewhat interesting.
Let’s speed-run the thing, Flash Gordon every n-IB from 1-IB to 1000-IB. Which ones converge, which ones don’t?
Golly, how poor was my paradigm of the nature of life, that on an arbitrarily long time scale (assuming your arbitration is bounded at 1000), mindless looping is the order of the day? 55.8% of board configurations end up looping, only 44.2% end up converging.
Granted, this is at a pretty high flight level. What if we bucketize them into equisized 10 groups (based on what “hundreds group” they are in, to put it lamely) for a little more granularity?
Hard to make much heads or tails of it really. As little as 45% of IB looping from 1 to 100*, and as many as 63% looping in 501-600, and 701-800. Vexing.
Now, if I change the grouping ever so lightly, to go by “tens” instead of hundreds, what might that look like?
*(I realize 100 should technically be in the “1” bucket, but since I started at 1 instead of 0, since I wanted equally sized buckets, and since whether or not a specific number converges / loops is not of particular interest to us here, so much as the behavior in the aggregate – for all these reasons IB at the very edge of bins are “rounded down”, so to speak. So in the next graph, 10 is in the 0 group, 100 is in the 9 group, et cetera.)
Yikes. Randomness galore. Seemingly.
This mangled sawtooth of a Ukranian flag doesn’t do much to dispel the impression of random noise (the color scheme for this dashboard, I should point out, was in fact the Colombian flag).
To see the metamorphosis from the dawn of life to “56-ish percent of Identity Boards smaller than 1000 converge”:
The slow incremental climb.
Let’s go at this the old Null Hypothesis – Alternate Hypothesis approach, treating the 1000 consecutive binary outcomes of looping (1) and converging (0) as a binary string. (Though actually, we’ll lop off the first 10, since those don’t quite have the patina of randomness; so 990).
If we have the binary string of length 990, we would expect a certain distribution of run lengths of consecutiveness, if in fact the string were truly random. Not unlike the old heads or tails metaphor, if that helps.
Flipping a coin 1000 times, we would expect
- 4 consecutive heads something like 15 different times
- 2 consecutive tails 61-ish different times
- 7 consecutive heads something like 2 times
And the like. All fun facts that we would expect to hold, if in fact this sequence was truly random.
Likewise with our Identity Board sequence (if our default assumption is that it is random)! So let’s apply the same logic. First, what is our run distribution?
(I.e. how many times did we have EXACTLY 5 consecutive looping boards, or EXACTLY 3 consecutive converging boards, et cetera).
Aha! By way of tutorial for reading this chart, one can say such silly things as:
“There were 76 times that boards converged exactly 2 consecutive times.”
OR
“There were 59 times that boards looped exactly 2 consecutive times.”
OR
“There were 108 times that boards converged, then looped, and had the streak abruptly broken with another convergence. Likewise, there were 128 ‘standalone’ converging instances.”
You get the idea.
What we further get a sense of here is that is looping runs tend to go on longer streaks, whereas converging runs have a greater tendency of being standalone 1-run or 2-run events. The precise quantification of that comes a little later, though suffice it so say for now that is a titillating observation.
How many consecutive runs of different lengths of looping and converging did we observe, relative to what we expected?
Okay, visually with this sideways azure skyscraper, you get the sense that the few big, “this exceeds our expectations, whereas this fails to meet our expectations” are:
- We get fewer 1-run looping boards than we expect.
- We get more 2-run converging boards than we expect.
- We get more 4-run and 5-run looping boards than we expect.
There is a way of making this a slight bit clearer, via the Chi-Squared statistic, and direct plotting of the deltas. That’s right, the tried and true, “do not pass go, do not complete the module until you have rounded to the nearest 4th decimal”-style statistical test.
So, plain Ye Olde English, what does this tell us?
- When Identity Boards loop, they are (relatively) more likely to do so 3 or 4 at a time.
- When Identity Boards converge, the behavior is close to random. Only kinda sorta, one can say they happen in pairs more than expected.
- Or, as we said above, looping runs are likely to be more clustered; converging runs tend to come solo, or in pairs, but are less likely to agglomerate to anything more than that.
The reason why … is surely worth investigating. In another lifetime perhaps.
A final visual to battering ram your occipital lobe: the grid of looping and convergence.
I can’t get it all in one screen grab (even Snippy is not omnipotent), but believe you me, it is on the dashboard. And what’s more, again the zoom resolution fails me, but this is another note of interest, re-packaged in tabular form:
A disproportionate quantity of numbers that Mod(x,10) = 2 (i.e., numbers with a “2” in the one’s place) wind up looping. Interesting.
But not interesting enough to extend this tinfoil theorizing any further. For now.