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All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey.
On the left, each row and each column is associated with a certain object or concept; there is a rule for combining rows and columns to give images; it would be possible to extend with new rows/columns or delete/reorder any existing columns.
On the right, this is not so. The rule might be about how the images must relate to their neighbors, for example.
Left examples are a generalized version of the analogy grids seen in BP361. Any analogy a : b :: c : d shown in a 2x2 grid will fit on the left here.
To word the solution with mathematical jargon, "defines a (simply described) map from the Cartesian product of two sets vs. not so." Another equivalent solution is "columns (alternatively, rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.
Similar idea that does not quite work as a solution here: any removed square could be reconstructed based on the rule vs. not.
There is a trivial way in which any example can be interpreted so that it fits on the left side: imagine each row is assigned the list of all the squares in that row and each column is assigned its number, counting from the left. But each grid has a clear rule that is simpler than this. | 
