As the name implies, Bitcoin puzzle transactions are created as a challenge and not for the usual purpose of monetary transfer.
Puzzles can be published for several reasons, including software test bounties, marketing campaigns or just for fun.
In the early days of Bitcoin, when $BTC coins could be found for free on the WWW, it was common to set up challenges involving large amounts of coins. (Even legitimate Bitcoin faucets were commonplace back when the monetary value of BTC was in the pennies range. “Free Bitcoin” was actually a thing!)
These early puzzles normally stemmed from technically curious groups, cryptographers, cypherpunks and others who wished to test Bitcoin’s resilience against distributed attacks.
As Bitcoin gained higher financial value, the “naive puzzles” of yonder slowly disappeared.
Bitcoin puzzle transactions are now usually part of some marketing campaign, such as for promoting a website or product.
Challenges with very high potential rewards have the potential to generate instant buzz, receive inbound WWW links and get loads of free attention.
Bitcoin puzzles are still an excellent marketing tool. A Google search for “Bitcoin puzzle” will yield recent news articles mentioning large unclaimed prizes. Some Bitcoin puzzles go back several years, such as this one mentioned on Bitcointalk.
Bitcoin puzzles can be controversial for several years.
For example, there is no way to verify that the winner isn’t the puzzle creator himself. Remember, Bitcoin addresses can be generated for free. Prize winners wouldn’t identify themselves, either for their personal security or other reasons. This would make a legitimate winning address just as valid as a fake one generated by the puzzle creator himself.
Another possibility is creating zero possible solution games. Basically, if a scammer is able to raise a high enough amount of BTC to grab the community’s attention, and then proceed to generate a phony puzzle with zero possible solutions (random numbers) with a high bounty placed on it, it becomes impossible to audit the process without having access to the private key which created the puzzle. Challengers could end up spending countless hours in vane, not knowing the puzzle doesn’t have possible solutions.
Some puzzles offer a signed message as proof of BTC ownership, which is fine for the purpose of proving that the funds do belong to (or have been under the control of) the puzzle creator. But there’s also no way to know if the winning address is owned by the puzzle owner himself.
Seeing how several puzzle addresses (see references) now display zero balances, it’s possible that the original BTC owners themselves withdrew the coins after Bitcoin gained significant value in recent years.