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Cambridge interview questions

Here is a math interview question that was kept in reserve for the 2021 Girton admissions. Can you solve it? Bonus points if you know how it applies to theoretical physics :)
Here is a hint: how does the superficial divergence of a \(L=\partial_\mu\phi\partial^\mu\phi/2-\lambda\phi^n/n!\) diagram go with loops, legs and vertices?