$\begin{array} { l l l l l l } & S & E & V & E & N \\ & S & E & V & E & N \\ +& & & S & I & X \\ \hline \\ T & W & E& N&T&Y\\ \end{array}$

Each letter represents a distinct digit.

What is the value of $W$ ?

2
5
9
4
7
8
3
6

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I actually solved it with coding using Python before I provide the solution above. My coding is as follows:

```
# 7+7+6=20
from itertools import permutations
for a in permutations("0123456789", 9):
S = a[0]
E = a[1]
V = a[2]
N = a[3]
I = a[4]
X = a[5]
T = a[6]
W = a[7]
Y = a[8]
if S != "0" and T != "0":
if 2*int(S+E+V+E+N) + int(S+I+X) == int(T+W+E+N+T+Y):
print S+E+V+E+N, S+I+X, T+W+E+N+T+Y
```

Chew-Seong Cheong
- 6 years, 4 months ago

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()
```

10 Helpful
3 Interesting
0 Brilliant
1 Confused

that was amazing

Caio Espínola
- 6 years, 4 months ago

using namespace std;

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<<" ";
}
}
}
}
}
}
}
}
}
```

}

4 Helpful
2 Interesting
1 Brilliant
0 Confused

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.

1 Helpful
0 Interesting
0 Brilliant
1 Confused

Can you add your code of how you used a random generator? This way, others can learn from it.

Note that a solution set need not be unique, and that we are also not guaranteed that a solution set exists.

Permutation is quite a tedious task that I have done it before for other question found else where which I called a runner. Random can make easy task with the speed and power to consider all for limited space with effective spans. Since you may like permutation as a better solution, I think I just keep my method for own use. The main idea is to set a list of arrays in order and then swap each and everyone by doing a one to one interchange. Having proper equations to add up what we want, when the computer got it, it ceases to acknowledge.

Lu Chee Ket
- 6 years, 4 months ago

I thought of randomising them until they're correct, but I then decided an ordered iteration will get it faster, with a maximum of only 1 million iterations.

Shadi Barghash
- 6 years, 4 months ago

On average, because a randomized algorithm could result in duplicate searches, it will be slower.

It ultimately depends on how what the solution set looks like. In this case, we don't really have much of a clue without doing any further analysis.

That what I thought of :)

Shadi Barghash
- 6 years, 4 months ago

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

1 Helpful
0 Interesting
0 Brilliant
1 Confused

How did you reach that conclusion?

IF A=1 ,B=2...Z=26 Then SEVEN=19 5 22 5 14 ,So FOR S=19+5=24=2+4=6,E=19+5+2=26=2+6,V=19+5+2+2=28=2+8=10=1+0=1,

E=19+5+2+2+5=33=3+3=6,N=19+5+2+2+5+14=47=4+7=11=1+1=2,

So SEVEN=68162,

Similarly=SIX=19 9 24,WHERE S=1+9=1,I=19+9=28=2+8=10=1+0=1,X=19+9+24=52=5+2=7,For this SIX=117

So 68162+68162+117=136441=TWENTY .So W=3

GOUTAM TRIPATHY
- 6 years, 4 months ago

According to you SEVEN = 68162 but shouldn't the second and second last digit (8 and 6 in you r solution) be same?

Ritvik agg
- 6 years, 4 months ago

I'm sorry, but I do not understand what you are trying to say. How did you go from "SIX = 19 9 25" to "where S=1+9=1, I=19+9=28=2+8=10=1+0=1, X=19+9+24=52=5+2=7"?

And if "SIX = 117", and "SEVEN = 68162", is the value of S equal to 1 or 6?

×

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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 $V+V+S+2 = 20 +N$ :

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

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

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

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