## 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 ${11}$.

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 ${n\approx 2.7}$.

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 ${n\approx 2.7}$.

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

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

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