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#### Problem

The cube, 41063625 (345^{3}), can be permuted to produce two other cubes: 56623104 (384^{3}) and 66430125 (405^{3}). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.

Find the smallest cube for which exactly five permutations of its digits are cube.

#### Solution

open System let cubeRoot (n:int64) = Math.Pow(double(n), 1.0/3.0) // define function to investigate the numbers of d digits which are cubes let f d = // find the min & max number whose cube is d digits long let min, max = int64(cubeRoot (pown 10L d)), int64(cubeRoot (pown 10L (d+1))) // and look for groups of 5 numbers which are cubes and permutations of one another [min..max] |> List.map (fun n -> pown n 3) |> Seq.groupBy (fun n -> n.ToString().ToCharArray() |> Array.sort) |> Seq.filter (fun (k, l) -> l |> Seq.length = 5) let answer = // go through the numbers of a given number of digits and find the first set of groups // of 5 numbers which are cubes and permutations of one another let groups = Seq.unfold (fun state -> Some(state, state+1)) 7 |> Seq.map f |> Seq.filter (fun l -> Seq.length l > 0) |> Seq.head // find the smallest elements in the groups groups |> Seq.map (fun (k, l) -> l |> Seq.min) |> Seq.min

Bryan SlatnerYour use of groupBy to find the permutations is *extremely* clever.

Yan Cuithank you :-)