On the finite set of missing geometric configurations (n4)(n4)

We present various methods that have reduced the finite set of missing geometric configurations (n4)(n4). We show that there is no geometric configuration (174)(174), and we provide examples for the former unknown cases n=18n=18, n=29n=29, n=31n=31 — there do exist geometric configurations (n4)(n4). To construct these we use methods from computer algebra and optimization.