Polylogarithms from diagrams with elliptic obstructions

Numerical evaluations of 2-loop kites and 3-loop tadpoles with several elliptic obstructions lead to remarkable empirical evaluations in terms of polylogarithms, for which proofs are very hard to find, notwithstanding intensive use of packages such as HyperInt and MZIteratedIntegral. I describe the efficient methods by which puzzlingly simple results were obtained and hopes for demystifying them.