# Programmatically solving a math puzzle with four unknowns and four equations

I have this puzzle and I want to solve by code. I wrote simple code trying to brute force but it fail in one condition

https://dotnetfiddle.net/cJyu3w How can I solve such problems?

• Try this.
– Jens
Mar 24 '20 at 2:04
• This one has no solution. Mar 24 '20 at 2:17
• @DanielMathias Or, at least, there is no integer solution. Mar 24 '20 at 2:45
• a= 3.5 , b= 4.5, c= 9.5, d= 3.5 but I guess that's not what you want. Mar 24 '20 at 2:46
• ^ this is the only solution tho: see the columns, we know the sum of 4 numbers is 13+8=21. As the sum of first row is 8, the sum of second row is 13. Solving x+y=13, x-y=6 leads only one single solution. Mar 24 '20 at 5:54

First of all, let's see why your brute-forcing fails. (This is the puzzle part, the rest is plain old math.)

Let's check the parity of the answer numbers, by choosing an odd number (first picture) or an even number (second picture) as the top left unknown:  No matter which you chose, the number at the bottom right would have to be both odd and even at the same time, so there's no integer solution.

However, there are four equations and four unknowns, so we should have at least one solution (unless the system is inconsistent), or even many, if the equations aren't independent.

Since you wanted to know how to solve this by a computer program, let's do this with Octave.

(Sometimes it's difficult to choose the best programming language for a given problem. Here, we have a free language that actually gives the solution to our problem as the first syntax example on its home page. Choices don't really come easier than that.)

To get started, let's write the four equations out. Let's name the unknowns with letters a-d, so we get:

  a +b       = 8
c -d = 6
a    +c    = 13
b    +d = 8


Notice that I've placed all the letters in their own columns. This is important for the next step, where we turn this into a matrix of coefficients. That is, we only write the multipliers (0, 1 and -1 in this case) into the columns, and put some brackets around the stuff:

$$\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 \end{bmatrix} \begin{bmatrix} a\\ b\\ c\\ d \end{bmatrix} = \begin{bmatrix} 8\\ 6\\ 13\\ 8 \end{bmatrix}$$

(The column with the letters is there just to keep the notation more or less mathematically valid.)

Then, we just type these in and let Octave solve it (using the backslash operator (A\b)):

A=[1 1 0 0
0 0 1 -1
1 0 1 0
0 1 0 1]

b=[8
6
13
8]

A\b


(You can Try it online!)

This gives us the answer for all the unknowns (or a warning about a singular matrix if our equations weren't independent, or if our system was inconsistent):

   3.5000
4.5000
9.5000
3.5000


Since we got a neat result without any warnings, we can just type it in the proper boxes to achieve the final result:

• bass has an excellent answer, but missed a quick simplification. The top row a + b = 8 and the right column b + d = 8 share a common addend and result. therefore, a = d. for a generalized solution, this might not be helpful, but for the question presented, it simplifies the problem.
– Jim
Mar 24 '20 at 17:26