Euler's number can be estimated by using an array of size $n$, randomly permuting it and checking whether it is a derangement or not. The simulation repeats this process several times and keeps track of how many times this permutation was not a derangement. This simulation does not require us to generate fractions.

This process is special in the sense that it uses integers to estimate an irrational number.

Is there a similar process to estimate the constant Pi, which uses only random integers or random permutations?

Let me illustrate what I DO NOT want. Pi can be estimated by picking random points in the square whose corner points are (-1,-1) (-1,1)(1,-1) and (1,1). We also draw a circle inside this square with radius 1 and origin as its center. Let $c$ be the number of points inside(or on) this circle and $s$ be the set of points which lie inside the square but outside the circle. $c/s$ approaches $\pi/4$. I do not want to use this method, because it requires me generate random fractional numbers in the range of -1 to 1, to generate these coordinates.

I hope my requirement is clear.

Euler's number can be estimated by using an array of size $n$, randomly permuting it and checking whether it is a derangement or not. The simulation repeats this process several times and keeps track of how many times this permutation was not a derangement. This simulation does not require us to generate fractions.

This process is special in the sense that it uses integers to estimate an irrational number.

Is there a similar process to estimate the constant Pi, which uses only random integers or random permutations?

Let me illustrate what I do not want. Pi can be estimated by picking random points in the square whose corner points are (-1,-1) (-1,1)(1,-1) and (1,1). We also draw a circle inside this square with radius 1 and origin as its center. Let $c$ be the number of points inside(or on) this circle and $s$ be the set of points which lie inside the square but outside the circle. $c/s$ approaches $\pi/4$. I do not want to use this method, because it requires me generate random fractional numbers in the range of -1 to 1, to generate these coordinates.

I hope my requirement is clear.

Euler's number can be estimated by using an array of size $n$, randomly permuting it and checking whether it is a derangement or not. The simulation repeats this process several times and keeps track of how many times this permutation was not a derangement. This simulation does not require us to generate fractions.

This process is special in the sense that it uses integers to estimate an irrational number.

Is there a similar process to estimate the constant Pi, which uses only random integers or random permutations?

Euler's number can be estimated by using an array of size $n$, randomly permuting it and checking whether it is a derangement or not. The simulation repeats this process several times and keeps track of how many times this permutation was not a derangement. This simulation does not require us to generate fractions.

This process is special in the sense that it uses integers to estimate an irrational number.

Is there a similar process to estimate the constant Pi, which uses only random integers or random permutations?

Let me illustrate what I DO NOT want. Pi can be estimated by picking random points in the square whose corner points are (-1,-1) (-1,1)(1,-1) and (1,1). We also draw a circle inside this square with radius 1 and origin as its center. Let $c$ be the number of points inside(or on) this circle and $s$ be the set of points which lie inside the square but outside the circle. $c/s$ approaches $\pi/4$. I do not want to use this method, because it requires me generate random fractional numbers in the range of -1 to 1, to generate these coordinates.

I hope my requirement is clear.