I had this funny feeling reading this, like "I've met these people before". Not literally the individual litigants or sore losers here, but people kind of like them.

Then I remembered -- as I've mentioned a number of times here (including recently on the thread about the Great Internet Mersenne Prime Search), I used to respond to https://www.eff.org/awards/coop e-mail. This is an award where EFF will give out monetary prizes for people who find large prime numbers.

A surprising number of people developed totally unshakeable confidence that they had won the award and were entitled to the money, and some of them were very persistent and very angry at me for telling them that they hadn't. Some of them wrote to me more than 80 times to try to convince me of the importance of their discoveries and to try to persuade me to award them the prize. (I'm actually kind of grateful after reading this article that none of the claimants ended up sending me a death threat.)

There were a couple of people who sent me their supposed "formulas for prime numbers" and developed a grandiose attachment to the importance of their supposed discoveries. Some of them would cc the Nobel Foundation (!) or the Clay Mathematics Institute (!) or the United Nations (!!) on their e-mails. Presumably they did this because they felt these discoveries were of such tremendous importance that they expected to receive multiple organizations' awards for them. (One or two people explicitly stated that, and suggested that their discoveries would also be important in resolving other unsolved problems in mathematics and physics.)

So one oddity about this (no pun intended) is that for some of the people, I would (probably against my better judgment and definitely against my colleagues' advice) send them specific refutations for their claims. For example, they might claim a specific number, and I would factor it with sympy, or they might claim all numbers of a specific form (e.g. "a prime number of 1s in a row"), or they might claim a formula, and I would, in addition to sending them a link to the Wikipedia article discussing how "formulas for primes" of certain forms can never exist, send them a specific counterexample. Like "sorry, 1111111111111 has thirteen 1s in a row, but it is not prime because it is divisible by 53".

Several of the people would look at my counterexamples, and, instead of concluding "oh, my reasoning must have been wrong because I was totally sure but my conclusion was mistaken" or "oh, apparently I haven't properly understood the concept of mathematical proof", they would send a new formula that was slightly tweaked, sometimes by adding 4 or something. If I then factored the resulting number, these people would then tweak the formula again and insist that they had fixed it.

At the time I was frustrated that it was so hard to get across the concept of deductive proof to people (the idea that we know that some formulas and algorithms and equalities are correct, for deductive reasons, and not just based on lucky guesses or conjecture). But thinking about this together with the Fenn treasure stuff, I want to say that there are some much deeper psychological things going on, about some people looking at a contest or treasure hunt of a certain kind and forming a resolve to be the one to solve it, leading to an unwarranted certainty that they have done so.

So I think this is a real hazard of offering a prize, especially for solving an alluring, romantic mystery that has resisted other people's efforts to resolve it: it seems that some people's identities quickly get caught up in a notion of "I'm The One" and really persistently flinch away from efforts to disabuse them of that idea.

It's also interesting to me to think how I'm involved in a lot of organized puzzle stuff where there are public challenges for people to race against each other to be the first one to resolve mysteries or challenges, but mostly those events seem to avoid these particular phenomena (not only do people not tend to threaten each other or dispute the outcome or whatever, but they tend to be very sporting about it, and consistently defer to the judgment of the puzzle creator in determining what counts as the right answer, although they might criticize a puzzle aesthetically as having too many ambiguities or red herrings or something). I guess we're lucky that these events attract people who are motivated to follow the rules and norms of the game. Maybe the puzzlehunt world benefits from (1) typically not having any prize except bragging rights or commemorative/sentimental objects, (2) having a time limit where the contest ends at some point, and (3) having a clearly identified author or editor who is recognized as having created the challenge and thereby having the right to decide what counts as a correct solution to it. (In fact, things like Cicada 3301, that lack (2) and (3), have reportedly caused some people to become kind of obsessed, not in a good way.)