When the digits of pi go to 11

There is an upward trend in the digits of {\pi}. I just found it using Maple.

X := [0, 1, 2, 3, 4, 5, 6, 7, 8]:
Y := [3, 1, 4, 1, 5, 9, 2, 6, 5]:
LinearFit([1, n], X, Y, n);

2.20000000000000+.450000000000000*n

Here the digits are enumerated beginning with the {0}th, which is {3}. The regression line {y = 2.2 + 0.45n} predicts that the {20}th digit of {\pi} is approximately {11}.

It goes to 11
It goes to 11

But maybe my data set is too small. Let’s throw in one more digit; that ought to be enough. Next digit turns out to be {3}, and this hurts my trend. The new regression line {y=2.67+0.27n} has smaller slope, and it crosses the old one at {n\approx 2.7}.

Next digit, not as good
Next digit, not as good

But we all know that {3} can be easily changed to {8}. The old “professor, you totaled the scores on my exam incorrectly” trick. Finding a moment when none of the {\pi}-obsessed people are looking, I change the decimal expansion of {\pi} to {3.1 41592658\dots}. New trend looks even better than the old: the regression line became steeper, and it crosses the old one at the point {n\approx 2.7}.

Much better!
Much better!

What, {2.7} again? Is this a coincidence? I try changing the {9}th digit to other numbers, and plot the resulting regression lines.

What is going on?
What is going on?

All intersect at the same spot. The hidden magic of {\pi} is uncovered.

(Thanks to Vincent Fatica for the idea of this post.)