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?

Here is another question which was kept in reserve for the 2022 admissions.

Finally another one from 2023. This question was created entirely by Stefan Ivanov.