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Alex,

Yes, there's not enough info here... It's laid out apparently as an open-ended question. That might have made a good oral exam question with a primary purpose of getting the Tripos "contestant" to "think on his feet". But I don't think the Cambridge Mathematical Tripos had anything like that, and a question on lunars would have seemed like a dinosaur fossil by 1872 and also much too easy.

I have in mind a modified, more detailed version of this Kelvin puzzle, which I will try to post tomorrow. It would be a similar "minimalist" setup but with enough details to give us something calculable. 

Is the puzzle even real? It may just be part of the joke -- a tall tale poking a little fun at Kelvin both for his late-in-life growing irrelevance and also his fascination with nautical matters. It also could be that the anonymous witness repeating this story was providing an account ten times removed... The story says it dated from the Tripos of 1872, but this "tribute" was published in 1908. That's like someone today, in 2026, writing about an exam question from 1990. 

Frank Reed