It would be really interesting to see if an AI is able to generate its own wallet and send the funds there.
I don't doubt this is possible if you create an AI agent who has deep access to a Large Language Model. However, the LLM itself would not be able imo because this is not what it was designed for. It is designed to react to human queries.
What would be needed is some kind of "fake consciousness". This looks like some ultra-advanced sci-fi but would be simply a routine regularly executing certain kinds of queries. It has to be trained to query the LLM to reach certain goals.
I think the current state of technology doesn't allow such an auto-query process covering most aspects of human life or even the field of a professional worker, as this would be very close to an "AGI" already. But a "specialist fake consciousness", e.g. a program all the time trying to create and improve ways to earn cryptocurrencies, would be probably possible.
One could for example start first with some kind of trading bot and then improve it giving it more information/context about ways how crypto can be earned. The next step would be to let him discover new ways to earn, and finally, auto-improve this "game" until it becomes good enough to solve Pellegrini's "puzzle".
It seems at first difficult to set apart such an AI from a traditionally programmed program (humans simply tell them how to search for the data). However, as LLMs are very large and complex, it can be difficult to find such an algorithm traditionally, so some form of AI training would probably be needed for this kind of program/agent.