$\pi$ = 13, $\sqrt{2}$ = 7, $e$ =?

Question

_{From The Escape Room Advent Calendar: Puzzle book for adults}

I know the answer to this puzzle.

I cannot for the life of me figure out why it is what it is.

Answer 1

I think the answer is

$e=11$

Indeed, looking at

the approximate values given in the book, we can see that there are mistakes in the given number.

$\pi=13$:

3.14159265358919… where actually $\pi$ is approximatively equal to 3.14159265358979… ; the 13th digit is wrong.

$\sqrt{2}=7$:

1.41421386… where actually $\sqrt{2}$ is approximatively equal to 1.41421356… ; the 7th digit is wrong.

So:

Since $e$ is approximatively equal to 2.718281828459…, the first wrong digit in 2.718281828419… is the 11th.