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A packing $k$-coloring is a natural variation on the standard notion of graph$k$-coloring, where vertices are assigned numbers from $\{1, \ldots, k\}$, andany two vertices assigned a common color $c \in \{1, \ldots, k\}$ need to be ata distance greater than $c$ (as opposed to $1$, in standard graph colorings).Despite a sequence of incremental work, determining the packing chromaticnumber of the infinite square grid has remained an open problem since itsintroduction in 2002. We culminate the search by proving this number to be 15.We achieve this result by improving the best-known method for this problem byroughly two orders of magnitude. The most important technique to boostperformance is a novel, surprisingly effective propositional encoding forpacking colorings. Additionally, we developed an alternative symmetry-breakingmethod. Since both new techniques are more complex than existing techniques forthis problem, a verified approach is required to trust them. We include bothtechniques in a proof of unsatisfiability, reducing the trusted core to thecorrectness of the direct encoding.

The Packing Chromatic Number of the Infinite Square Grid is 15