Let $c_n = 0,1,2$ be the carried-forward. From $S+S+c_4= TW$ , it is clear that $T = 1$ or $2$ .

If $T=2$ , then $S=9$ , $c_4=2$ , $E=8$ and $W=0$ . From $E+E+I+c_1=10c_2 +T \Rightarrow I+c_1 = 6$ and $c_2 = 2$ . There are three cases for $V+V+S+2 = 20 +N$ :

• $V=5$ and $N=1$ , then $N+N+X = Y\Rightarrow X=0=W$ , which in not acceptable.
• $V=6$ and $N=3$ , then $N+N+X = Y\Rightarrow X=6=V$ , not acceptable.
• $V=7$ and $N=5$ , then $N+N+X = Y\Rightarrow X=2=T$ , not acceptable.

Therefore, $\boxed{T=1}$ .

Now, from $E+E+I+c_1 = 10c_2 + T = 1,11,21$ .

• If $c_2 = 0$ , the only option is $E=0$ and $I = 1 = T$ which is not acceptable.
• If $c_2 = 1$ , there are two options:
• $\quad \Rightarrow E=5$ and $I=1=T$ , not acceptable;
• $\quad \Rightarrow E=4$ and $I + c_1 = 3$ , but then it is impossible to have $\quad \quad \space E+E+c_3 = 10c_4 + E$
• Therefore, $c_2 = 2$ and there are two options:
• $\quad \Rightarrow E=7$ , $I =6$ and $c_1=1$ , again it is impossible to have $\quad \quad \space E+E+c_3 = 10c_4 + E$
• $\quad \Rightarrow$ the right option is $\boxed{E=8}$ , $I + c_1 =5$ , $c_3 =2$ and $c_4 = 1$

From $V+V+S+2=20+N$ , we have three options:

• $V=9$ , then $S=N$ , not acceptable.
• $V=6$ , $S=7$ and $N=1=T$ , not acceptable.
• Therefore, $\boxed{V=7}$ , $\boxed{S=6}$ and $\boxed{N=2}$ .

From $S+S+c_4= TW \quad \Rightarrow 6+6+1 = 13 \Rightarrow \boxed{W=3}$

I first tried to solve it by hand, then I decided to write a code to do it for me.. and it did! :D It iterates every letter for all possible digits while also making sure it's not assigned to different letter. Here it is:

Dim T As Byte = 0
    Dim W As Byte = 0
    Dim Y As Byte = 0

    For N As Byte = 0 To 9 Step 1
        For X As Byte = 0 To 9 Step 1

            If X <> N Then

                For E As Byte = 0 To 9 Step 1

                    If E <> N And E <> X Then

                        For I As Byte = 0 To 9

                            If I <> N And I <> X And I <> E Then

                                For V As Byte = 0 To 9 Step 1

                                    If V <> I And V <> E And V <> X And V <> N Then

                                        For S As Byte = 0 To 9 Step 1

                                            If S <> V And S <> I And S <> E And S <> X And S <> N Then

                                                Dim SEVEN As String = String.Format("{0}{1}{2}{3}{4}", S, E, V, E, N)
                                                Dim SIX As String = String.Format("{0}{1}{2}", S, I, X)

                                                Dim theSum As Integer = (CInt(SEVEN) * 2) + CInt(SIX)

                                                Dim sumChars() As Char = theSum.ToString().ToCharArray()

                                                If sumChars.Length = 6 Then

                                                    T = CByte(Val(sumChars(0)))
                                                    W = CByte(Val(sumChars(1)))
                                                    Y = CByte(Val(sumChars(5)))

                                                    If CByte(Val(sumChars(4))) = T And T <> S And T <> E And T <> V And T <> N And T <> I And T <> X And T <> W And T <> Y Then

                                                        If CByte(Val(sumChars(2))) = E And E <> W And E <> Y Then

                                                            If CByte(Val(sumChars(3))) = N And N <> W And N <> Y Then

                                                                If W <> S And W <> V And W <> I And W <> X And W <> Y Then

                                                                    If Y <> S And Y <> V And Y <> I And Y <> X Then

                                                                        Console.WriteLine(SEVEN)
                                                                        Console.WriteLine(SEVEN)

                                                                        Console.WriteLine(SIX)

                                                                        Console.WriteLine("=====")

                                                                        Console.WriteLine("{0}{1}{2}{3}{4}{5}", T, W, E, N, T, Y)

                                                                        Console.WriteLine()

                                                                    End If

                                                                End If

                                                            End If

                                                        End If

                                                    End If

                                                End If

                                            End If

                                        Next

                                    End If

                                Next

                            End If

                        Next

                    End If

                Next

            End If

        Next
    Next

    Console.ReadLine()

include<bits/stdc++.h>

using namespace std;

define f(i,n) for(int i=0;i<=9;i++){

int main(){

f(s,10)

    f(e,s)

        f(v,e)

            f(n,v)

                f(i,n)

                    f(x,i)

                        f(y,x)

                            f(t,y)

                                f(w,t)


        if(((s*10000+ e * 1000 + v * 100 + e * 10 + n)*2 + s*100 + i*10 + x)==t*100000 + w*10000 + e*1000 + n*100 + t*10 + y&&s!=e&&s!=v&&s!=n&&s!=i&&s!=x&&s!=t&&s!=w&&s!=y&&e!=v&&e!=n&&e!=i&&e!=x&&e!=t&&e!=w&&e!=y&&v!=n&&v!=i&&v!=x&&v!=t&&v!=w&&v!=y&&n!=i&&n!=x&&n!=t&&n!=y&&n!=w&&i!=x&&i!=t&&i!=y&&i!=w&&x!=t&&x!=w&&x!=y&&t!=w&&t!=y&&w!=y)cout<<w<<" ";



                                }
                            }
                        }
                    }
                }
            }
        }
    }
}

}

Most efficient permutation by random generator gives answer for W as 3

68782

68782

650

138214

S = 6

E = 8

V = 7

N = 2

I = 5

X = 0

T = 1

W = 3

Y = 4

With maximum of about 10 million or less than 0.5 million of trials, the answer must be found. An only answer was every time and it should be sole answer.

Seven + seven + six=twenty Then 68782+68782+650=138214 So, w=3