ant can be pruned from the domain of 1-across using the arc <1-across,1-down> as there is no element of the domain of 1-down that starts with "a".
Variable Domain 1-across {bus, has} 3-across {lane, year} 4-across {ant, car} 1-down {buys, hold} 2-down {search, syntax}
Here is one possible sequence:
This reaches a foothill, and never solves the goal.
1-across 3-across 4-across 1-down 2-down h-value ant book ant book browns 1 big book ant book browns 2 big book ant book ginger 2 big book ant buys ginger 2 big book big buys ginger 2 big buys big buys ginger 2 big hold big buys ginger 2 big lane big buys ginger 2 big lane big hold ginger 3
Yes. In fact it seems stupid not to do this. This works much better, if only because the search space is smaller, the plateaus seem smaller, and it more quickly finds better foothills. For example:
1-across 3-across 4-across 1-down 2-down h-value bus lane ant buys search 2 bus lane ant buys syntax 4 bus lane ant hold syntax 4 has lane ant hold syntax 5
This again works better, but the cost of each iteration (finding the neighbours) is more. It isn't obvious that the extra time would be better spent trying different starting points.
Here is one possible sequence:
1-across 3-across 4-across 1-down 2-down h-value ant book ant book browns 1 big book ant book browns 2 big book ant buys browns 2 big year ant buys browns 2 big year ant buys ginger 3 big year ant buys search 3 bus year ant buys search 4 bus year car buys search 5