Here's a variation on my previous post - another piece of hand-waving probabilistic reasoning, which I think is basically correct, but I suspect many will disagree.
A long time ago I used to do the challenging weekly cryptic crosswords in The Listener magazine. (The Listener has been defunct for many years, though I think the crossword continues in The Times.) The story was that if no-one solved the crossword, it was too hard, and if more than one person solved it, it was too easy. While that was an exaggeration, it wasn't easy, and I judged it worthwhile, if I completed the puzzle, to submit my entry for the prize draw.
I had completed about fifteen puzzles in a row, and submitted my answers each time, but hadn't won the book token. Then there was an unusual puzzle - it was mathematical rather than word-based. Two mathematician colleagues and I worked on it - we didn't find it at all easy - and we eventually solved it. I submitted our answer, and this time, we won the book token.
So - my conclusion was that (probably) more people solved the word-based puzzles than the mathematical one, and that therefore I was more likely to win the prize for that puzzle (as I had done) than for the others.
Is this conclusion valid?
As it happens, at the end of the year statistics for all The Listener crossword entries were published, so I was able to see if this was indeed the case. It turned out that the mathematical crossword had attracted about three times as many correct entries as any of the others. So my conclusion was in fact false, but I still think the reasoning was sound.
