It is a very interesting question as to what the large-scale structure of decomposition numbers for symmetric groups looks like. Does order or chaos reign? One way to approximate an answer to this question is to compute many examples, and see if one can see any patterns. Over many years, we (Elias, Libedinsky, Riche, Jensen,…) developed a way of computing much further out than was previously possible. I then spent a year implementing this and running it on a high-powered computer for 8 months, which led to the billiards conjecture (with Lusztig) in the 3-row case. One sees a beautiful pattern in a region (“billiards”) but we couldn’t guess what happens beyond. Interestingly, my implementation suffers from some unexpected difficulties, which blocks further progress. In more recent work with Riche (and inspired by work of Hazi) we worked out a way that these difficulties can be circumvented in principle. I’ll try to outline this new approach. Now all that remains is to actually implement it … I’ll report back in four years time.