Plants grown in large numbers for commercial purposes are usually set out on a regular grid, triangular, square or hexagonal, or possibly a rectangular grid. We need to understand their behaviour when they compete with each other for light, water and nutrients. In a greenhouse, there may be a lamp over each plant, half the plants having their lamps on and half off. The intensity of light falling on any particular plant is determined mainly by whether its own lamp is on, next by the number of its nearest neighbours whose lamps are on, and perhaps also by the number of its second-nearest neighbours whose lamps are on.
Such arrays are also used to study competition among different types of plants and, with some extra restrictions, to design field layouts as well. The problem of constructing suitable arrays was introduced in [CORMACK, R. M.: Spatial aspects of competition beween individuals. In: Spatial and Temporal Analysis in Ecology (R. M. Cormack and J. K. Ord, eds.), International Co-operative Publishing House, Fairland, Maryland, USA, 1979] and discussed in [GATES, D. J.: Competition between two types of plants with specified neighbour configurations, Math. Biosci. 48 (1980), 195–209]. Here we consider only the square grid.