no computers - French alphametic (corrected)

This is the corrected version of puzzle French alphametic (misspelled)

Every letter stands for a digit in base-10 representation, different letters stand for different digits, and the four summands and the sum are even:

         UN 
         UN
       DEUX
   +  DOUZE
  ------------

      SEIZE

Which digit does each letter represent?
(Please present the full analysis how these digits can be determined.)

Answer

Denote by $c_1$ the carry-over from the rightmost column, by $c_2$ the carry-over from the next column, by $c_3$ the carry-over from the middle column, and by $c_4$ the carry-over from the fourth column.

Since all four summands and the sum are even, the equation $2N+X+E=10c_1+E$ must be solved with $N,X,E$ even. By distinguishing five cases on $N$, we get the following five possibilities:

       N  X  c1 
     ------------
       0  0  0     (Case 0)

       2  6  1     (Case 1)
       4  2  1     (Case 2)
       6  8  2     (Case 3)
       8  4  2     (Case 4)

Case 0 has $N=X$, and therefore is illegal.
The next column in the summation yields $3U+Z+c_1=10c_2+Z$, and hence $U=(10c_2-c_1)/3$. Since $0\le c_2\le3$, this yields the following:

       N  X  c1 U  c2
     -----------------
       2  6  1  3  1  (Case 1)

       4  2  1  3  1  (Case 2)
       6  8  2  6  2  (Case 3)
       8  4  2  6  2  (Case 4)

Now Case 3 has $N=U$, and is illegal.
The middle column in the summation yields $E+U+c_2=10c_3+I$.

• 
  

  In Cases 1 and 2, this equation becomes $E+4=10c_3+I$. Hence $I$ is also even. The five subcases $E=0,2,4,6,8$ yield $I=4,6,8,0,2$. Since $E$ and $I$ must be different from $N$ and $X$, this leaves only the following subcases alive:

  
  

     N  X  c1 U  c2 E  I  c3
  
   --------------------------
     2  6  1  3  1  0  4  0   (Case 1a)
     2  6  1  3  1  4  8  1   (Case 1b) 
     4  2  1  3  1  6  0  1   (Case 2 )
  



• 
  

  In Case 4, the equation becomes $E+8=10c_3+I$. Hence $I$ is also even. The five subcases $E=0,2,4,6,8$ yield $I=8,0,2,4,6$. Since $E$ and $I$ must be different from $N,X,U$, this leaves only the following subcase alive:

  
  

     N  X  c1 U  c2 E  I  c3
   --------------------------
     8  4  2  6  2  2  0  1   (Case 4 )
  
  

Now let us turn to the fourth column, which yields $D+O+c_3=10c_4+E$. Since $E$ is even, this means that $D+O+c_3$ is even.

• In Case 4, the five even digits 0,2,4,6,8 ahave been assigned to $I,E,X,U,N$, respectively. Hence $D$ and $O$ both must be odd, which together with $c_3=1$ yields a contradiction.


• Similarly in Case 2, the only unassigned even digit is $8$. In this case the equation $D+O+c_3=10c_4+E$ becomes $D+O=10c_4+5$, which forces $c_4=1$ and $\{D,O\}=\{7,8\}$.


• Similarly in Case 1a, the only unassigned even digit is $8$. In this case the equation $D+O+c_3=10c_4+E$ becomes $D+O=10c_4$, which forces $c_4=1$ and $D+O=10$ with $O$ and $D$ odd and distinct.


• Finally in Case 1b, the only unassigned even digit is $0$. In this case the equation $D+O+c_3=10c_4+E$ becomes $D+O=10c_4+3$, which forces $c_4=1$ and $\{D,O\}=\{0,3\}$.

In all surviving cases, we have $U=3$ and $c_4=1$. In particular, this implies $S=D+1$, so that one of $S$ and $D$ is even.

• In Case 1a, we conclude $S=8$, $D=7$ and $O=3$; this subcase has $U=O$ and yields a contradiction.


• In Case 1b, we conclude $D=0$ and $S=1$, which has $D=0$ as starting digit of a number and hence violates one of the basic principles of alphametic puzzles.


• In Case 2, we finally conclude $D=8$ and $S=9$, and $O=7$.

Let us summarize:

       N  X  c1 U  c2 E  I  c3 D  S  O
     -----------------------------------
       4  2  1  3  1  6  0  1  8  9  7   (Case 2 )

Hence $I=0$, $X=2$, $U=3$, $N=4$, $E=6$, $O=7$, $D=8$ and $S=9$ have been assigned, and only the digits $1$ and $5$ remain open, and both can be legally assigned to $Z$.

This yields the following two solutions (that only differ in the value of $Z$):

     34               34
     34               34
   8632     and     8632 
+ 87316          + 87356
 ------           ------
  96016            96056