The Colorful Grid Puzzle: Finding the Missing Number in a Rainbow of Digits

Introduction

I found this problem as part of the Haselbauer-Dickheiser Test.

Problem

Determine the missing number:

Solution

We need to figure out how colors affect the numbers inside the white cells. Since we have no other information, we have to come up with sensible theories and test them out. A possible general one could be:

Colors represent a mathematical function applied to the number inside the corresponding cell, producing new numbers that are then added together to produce the result in the corresponding white cell.

Now the obvious question is: what function? Some possibilities are:

1. Adding a constant
2. Multiplying by a constant
3. Elevating the number to a constant

And many more. However, if colors do indeed represent a function, it's likely that it isn't much more complex than this, or otherwise the problem would be extremely hard. So let's try to analyze these simple ones first.

Because we have different colors, it's likely that the colors represent different mathematical functions, different constants, or maybe even both, but that would make things quite a bit more complex. Let's assume for simplicity's sake that they all represent additions but they each add a different constant. Let's name these constants $R$ for red, $G$ for green, $B$ for blue, $Y$ for yellow and $M$ for magenta.

Now let's analyze the first row of the grid:

If our theory is correct, adding all numbers and all constants together should result in 80:

In other words, the sum of all the colors should equal the value in the white cell minus the sum of all the numbers:

Let's simplify our grid by applying this theory:

If our theory is correct, all rows and columns that share the same frequencies for each color should have the same value on the corresponding white cell. You can verify that this is indeed the case. This is a very good sign that our theory might actually be correct, as it seems unlikely that this necessary condition would be satisfied if it weren't.

So now we have to figure out the constants for each color. This can be done by selecting a subset of 5 non-equivalent rows and columns and solving a system of equations. There are 3 non-equivalent rows and columns that don't have magenta, and 2 non-equivalent columns that don't have yellow. Let's choose these to be part of our system, as they will produce equations that only involve 4 unknowns each, hopefully making the process of solving the resulting system of equations easier:

Using equations $(1)$, $(2)$ and $(3)$ we can produce a smaller system of 2 equations without $Y$:

In a similar fashion, we now produce one equation without $R$:

Similarly, we use equations $(4)$ and $(5)$ to produce an equation without $M$:

Now, using $(6)$ and $(9)$ we produce another equation without $R$:

And from $(8)$ and $(10)$ we get $G=2$.

Now, with repeated substitutions, we end up with:

We then proceed to verify that our theory works with these constants in all rows and all columns of the grid, which indeed it does. All that rests now is to calculate the missing number:

The missing number is $112$.