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In the first part of the talk I will give a gentle introduction to finite projective planes and their generalization, combinatorial designs I will introduce mosaics of designs and give a short overview of the topic. I will then focus on a particular case where existence is an open problem: mosaics of projective planes of order four. The deceptively simple-looking problem is to find a 21 by 21 matrix with zeros on the diagonal and entries 1, 2, 3 and 4 elsewhere. Each nonzero entry should appear five times in every row and column and exactly once in the same position for every pair of rows or columns. I will talk about attempts to construct such an object by computer and seek advice from the MODUS audience on how to approach the problem computationally.