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This Python program tries to fit the dominoes into the grid recursively, checking the products as rows and columns are completed. It runs in 3.2s.

Solution:The four horizontal dominoes are: 1-2, 2-2, 3-4 and 4-4.Here is a diagram of the layout of the dominoes (horizontal dominoes are shown in red, vertical dominoes in blue):

There are two other ways to lay out the dominoes which give the same numerical grid, but each of them has six horizontal dominoes: