Bitcoin puzzle transactions are created as a challenge and not for the usual purpose of monetary transfer.
In the early days of Bitcoin, with the extremely low value per $BTC, it was common to set up challenges with large amounts. (Even legitimate Bitcoin faucets were commonplace back when the monetary value of BTC was in the pennies.)
These early puzzles normally stemmed from technically curious groups, cryptographers, cypherpunks and so on.
As Bitcoin gained financial relevance, these “naive puzzles” 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 instantly generate buzz, receive inbound WWW links and get loads of free attention. It’s an excellent marketing tool.
Bitcoin puzzles are controversial because there is no way to verify that the winner isn’t the puzzle creator himself.
Basically, if a scammer is able to raise a high enough amount of BTC and then generate a phony puzzle with zero possible solutions, but with a high bounty placed on it, it becomes impossible to audit without access to the private key which created the puzzle.
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 no way to know that the redeeming address is somehow linked to the puzzle owner.
Seeing how several puzzle addresses (see references) now display zero balances, it’s possible that the original BTC owners themselves withdrew the coins.