# The 7-Digits Puzzle (Puzzle7)

### Problem

Use each digit from 1 to 7 exactly once, and place them into the circles of Figure 1 in such a way that the sum along each of the five lines is the same. Figure 1: The 7-digit Puzzle

### Modeling Steps

The key observations are: There are 7 circles and 7 digits. On each line we have 3 circles, hence 3 digits to place. Which digit should be at the center? Or: what numbers should be at the two ends of a line? Once answered these questions, it is easy to see the solution. Two observations lead to the number in the center: (1) The sum must be even, since otherwise at least four lines must have two even and one odd numbers, but there are only three numbers in {1,, 7} that are even. (2) the sum of all numbers is 28. The sum on the three lines through the center is 28 + 2D and it must be divisible by 3. Since the sum is even, 28 + 2D can only be 36. Hence the sum is 12 and D = 4.

Let’s now formulate this problem as a mathematical model: Seven integer variables xi ∈{1,, 7} with i,j I = {A,, G} are to be determined and all must be different from each other. Furthermore, an additional variable z is introduced for the sum on each line.

Note that: Depending on the notation of the formulation, the sentence “all must be different from each other” could directly be interpreted as a predicate in the modeling formulation (as done here) or be translated as a set of inequalities as follows: In the LPL language the predicate is directly integrated as a “global constraint” with the keyword alldiff (another way in LPL is to use the function Alldiff()).

For each of the five straight lines a constraint is specified and the whole model can be formulated as follows : Listing 1: The Complete Model implemented in LPL 
```model Puzzle7 "The 7-Digits Puzzle";
set i,j := ['A','B','C','D','E','F','G'];
alldiff variable x{i} [1..7];
variable z;
constraint
L1: x['A'] + x['D'] + x['G'] = z;
L2: x['A'] + x['C'] + x['F'] = z;
L3: x['B'] + x['E'] + x['G'] = z;
L4: x['B'] + x['D'] + x['F'] = z;
L5: x['C'] + x['D'] + x['E'] = z;
solve;
Write('%s , z=%d' n',{i} Format('%s=%d ', i, x), z));
--draw the solution
parameter X{i}:=[10 50 10 30 50 10 50];
Y{i}:=[10 10 50 50 50 90 90];
set li{i,j}:=[('A','F') ('A','G') ('B','F') ('B','G') ('C','E')];
Draw.Scale(10,5);
Draw.DefFont('Verdana',20,0,2);
{li[i,j]} Draw.Line(X[i],Y[i],X[j],Y[j]);
{i} Draw.Circle(i&'='&x,X,Y,3,1,0);
--{i} Draw.Circle(i&'',X,Y,3,1,0);
end```

### Solution

The solution is given in Figure 2. The sum on each line is z = 12. Figure 2: The Solution to the 7-digit Puzzle

### References

   MatMod. Homepage for Learning Mathematical Modeling :  https://matmod.ch.

   Hürlimann T. Reference Manual for the LPL Modeling Language, most recent version. https://matmod.ch/lpl/doc/manual.pdf.