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In the square jigsaw puzzle problem one is required to reconstruct the complete image from a set of non-overlapping, unordered, square puzzle parts. Here we propose a fully automatic solver for this problem, where unlike some previous work, it assumes no clues regarding parts' location and requires no prior knowledge about the original image or its simplified (e.g., lower resolution) versions. To do so, we introduce a greedy solver which combines both informed piece placement and rearrangement of puzzle segments to find the final solution. Among our other contributions are new compatibility metrics which better predict the chances of two given parts to be neighbors, and a novel estimation measure which evaluates the quality of puzzle solutions without the need for ground-truth information. Incorporating these contributions, our approach facilitates solutions that surpass state-of-the-art solvers on puzzles of size larger than ever attempted before.