  We study the mixing time of a systematic scan Markov chain for sampling from
the uniform distribution on proper 7-colourings of a finite rectangular
sub-grid of the infinite square lattice, the grid. A systematic scan Markov
chain cycles through finite-size subsets of vertices in a deterministic order
and updates the colours assigned to the vertices of each subset. The systematic
scan Markov chain that we present cycles through subsets consisting of 2x2
sub-grids and updates the colours assigned to the vertices using a procedure
known as heat-bath. We give a computer-assisted proof that this systematic scan
Markov chain mixes in O(log n) scans, where n is the size of the rectangular
sub-grid. We make use of a heuristic to compute required couplings of
colourings of 2x2 sub-grids. This is the first time the mixing time of a
systematic scan Markov chain on the grid has been shown to mix for less than 8
colours. We also give partial results that underline the challenges of proving
rapid mixing of a systematic scan Markov chain for sampling 6-colourings of the
grid by considering 2x3 and 3x3 sub-grids.
