Solving https://mikesmathpage.wordpress.com/2016/04/29/a-strange-problem-i-overheard-bjorn-poonen-discussing/ with fs

ndimensionalpoo.fs

// Learn more about F# at http://fsharp.org
// See the 'F# Tutorial' project for more help.
open MathNet.Numerics
open System.Numerics

//create a fractional approximation of pi
let rec series n q = 
    seq { 
      yield q * 4N / n
      yield! series (n + 2N) (q * -1N) }
  
let pi = series 1N 1N |> Seq.take 100 |> Seq.reduce (+)
 
printf "pi: %s \n\n" (pi.ToString())


//caluclate a sqrt for BigRationals
let BrSqrt (x:BigRational) =     
    if x.IsOne || x.IsZero then
        x
    else
        let mutable temp = BigRational.FromInt(1)
        let precision = 100
        while temp.Numerator.ToString().Length < precision do
            temp <- ((x / temp) + temp) / BigRational.FromInt(2)
        temp


let rec factorial n = 
    match n with 
    | 0 -> BigInteger(1)
    | 1 -> BigInteger(1)
    | _ -> BigInteger(n) * factorial (n - 1)


let rec doubleFactorial n =    
    if n % 2 = 0 then
        match n with
        | 0 -> BigInteger(1)
        | 2 -> BigInteger(2)
        | _ ->  BigInteger(n) * doubleFactorial (n - 2) 
    else
        match n with 
        | 1 -> BigInteger(1)
        | _ -> BigInteger(n) * doubleFactorial (n - 2)
    
//get volume of n-ball
let volumeSphere radius dimension =    
    if dimension % 2 = 0 then
        let ik = dimension/2        
        
        let result = (BigRational.Pow(pi,ik) / BigRational.FromBigInt(factorial ik)) * BigRational.Pow(radius,2*ik)
        result
       // ((pi ** (float)k) / (float)(factorial k)) * (radius ** (float)(2 * k))
        
    else
        let ik = (dimension - 1) / 2
        let k = BigRational.FromInt((dimension - 1) / 2)
        let twok1 =2*ik+1
        let two = BigRational.FromInt( 2)
        let one = BigRational.FromInt(1)
        let kp1  = ik+1
        
        let top = (BigRational.Pow (two,kp1)) * (BigRational.Pow (pi, ik)  )
        let bottom = BigRational.FromBigInt( doubleFactorial twok1)
        let r = top / bottom        
        let result = r * (BigRational.Pow(radius,twok1))
        result
        
// hypercube volume
let volumeCube length dimension =
    BigRational.Pow(length, dimension)

// hypercube diagonal
let diagonalOfCube length dimension =
    length * BrSqrt(dimension)

[<EntryPoint>]
let main argv = 
    //dimensions to check
    [ 1205 .. 1207 ]    
    |> List.iter(fun e -> 
        //volume of smaller cubes
        let vcube = volumeCube (BigRational.FromInt(1)) e
        //colume of outter spheres
        let vsphere = volumeSphere (BigRational.Parse("1/2")) e
        //diagonal of smaller cubes
        let diagcube = diagonalOfCube (BigRational.FromInt(1)) (BigRational.FromInt(e))
        //inner cricle radius
        let innercircleradius = (diagcube - BigRational.FromInt(1)) / BigRational.FromInt(2)
        //inner circle volume
        let innercirclevolume = volumeSphere innercircleradius e
        //large cube volume
        let bigcubev = volumeCube (BigRational.FromInt(2)) e
        //diff between circle and cube
        let diff = innercirclevolume - bigcubev        
        let vsphereint = BigRational.ToBigInt(vsphere)
        let diffint = BigRational.ToBigInt(diff)
        let innervint = BigRational.ToBigInt(innercirclevolume)
        let radint = BigRational.ToBigInt(innercircleradius)        
        
        printf "dim: %i " e
        printf "incircrad: %s " (BigRational.ToBigInt(innercircleradius).ToString())
        printf "bigcube: %s " (BigRational.ToBigInt(bigcubev).ToString())
        printf "inv: %s " (innervint.ToString())
        printf "diff: %s\n" (diffint.ToString())
        )
        
   
    System.Console.ReadLine() |> ignore
    1