There's a 4 by 4 grid and a symbol in each box. Each symbol consists of a letter (A, B, C, or D) and a combination of a few shapes. Deduce which symbol is missing. Four options are presented.
One potential explanation:
Every letter must be at some point surrounded by a square and a circle (not necessarily at the same time).
The answer is:
Option 3 as B so far lacks a square
The count of distinct things in a row/column adds up to 7 (so a different letter or style counts as 1 but only once in a row)
So if we apply that to the row and column our questionable tile is on:
Row 3 has D,A,circle,line and box - five distinct things. Column 1 has D,C,B,box and circle - again five distinct things.
We find that none of those answers fit (I think) because they're repeated. Option 1 gives no new options to the row and only adds one to the column. 2 adds 1 to row and none to column. 3 gives one to row and one to column. 4 gives one to row and one to column. Either way we don't get the extra two we need. The answer would have to be something like B,triangle, line for this to work. Ah well...back to the drawing board. I was so convinced every row and column adding to 7 was relevant...
A potential rule:
We have letters and styles, each row/column either repeats a letter or repeats a style but won't repeat both.
Row 3 has two Ds and an A so we need a C or B without one of the other styles - The B in a box fits this. This also fits for column 1
I think the answer is
option 4: D inside a triangle.
Here is one possible way of approaching this puzzle:
Assign each letter and shape a number so that a slot is assigned a sum of letters and shapes. Figure out an assignment, so that filling a missing slot will turn the 4x4 square into kind of a semi-magic square: all the rows and all the columns will have same sum.
I made the following assignments
A = 8, B = 2, C = 4, D = -3
Circle = 3, Square = 1, Slash = -5, Triangle = 9
Option 1 becomes 12;
Option 2 becomes 7;
Option 3 becomes -2;
Option 4 becomes 6;
Now the square turns into:
7 9 -8 5
2 5 13 -7
6 -5 11 1
-2 4 -3 14
Option 4 is the only choice to make a puzzle into a semi-magic square.
I came up with this assignment by solving a very simple system of linear equations. The set of solutions is parameterized by letter B and a square shape :-)