That is not a logic puzzle [1], even though the author claims it is; it's a silly outcome that is easily achieved when one breaks how logic puzzles are presented and solved.
It's more of a bait-and-switch.
Here is a famous logic puzzle, often called a Knights and Knaves style puzzle [2]: Knights always tell the truth, Knaves always lie.
"John and Bill are standing at a fork in the road. John is standing in front of the left road, and Bill is standing in front of the right road. One of them is a knight and the other a knave, but you don't know which. You also know that one road leads to Death, and the other leads to Freedom. By asking one yes–no question, can you determine the road to Freedom?"
Here's a "solution" in the style of the author: "There is no solution, because the road to 'Freedom' is under construction, and the detour leads through 'Death.' Hahaha! Gotcha!"
It's a non-solution, and not a logic puzzle any more.
But the difference is that in this puzzle you can deduce that something is wrong by just using logic: In the absence of any hints, you'd just consider everything true. This obviously does not lead to a solution, as the red box states that one box lies. When you come to this point, you might wonder why you were trusting the boxes in the first place and reconsider your assumptions, i.e. not putting any trust in what the boxes say. After this step you realise that there suddenly is not enough information to determine which box the treasure is in.
The difference between the blog post and your example is that in your example there's nothing that makes you recheck your assumptions. A puzzle in the style of the author's would be:
Knights always tell the truth, Knaves always lie.
John and Bill are standing at a fork in the road. John is standing in front of the left road, and Bill is standing in front of the right road. John says: "One of us is a knight and the other a knave". You also know that one road leads to Death, and the other leads to Freedom. Bill says: "My road leads to freedom".
If you attacked that formulation with the normal assumptions for logic problems, the answer would be the same. If you break the assumption that you can trust all logical statements in the problem, regardless of their origin, then you could get a solution like "John is a dick and tells a mixture of truth and lies as he sees fit just to mess with you". But you could get that solution with any problem where the source of information is an entity in the problem itself, which is why we tend to assume that isn't the case.
That's entirely wrong. The whole point of the article is that there are two distinct types of information present in all problems of this type: information about the problem (which the author calls certification, and which is assumed to be true), and then information presented in the problem, which - in almost every problem I've ever seen of this type - has statements that aren't true in it.
> "If you break the assumption that you can trust all
> logical statements in the problem"
That is never the assumption - you are normally told explicitly that you can trust some and not others.
I disagree. In your case, the problem formulation is inconsistent with the answer (actually I don't think one question in this case suffices anyway): from the formulation we expect that the roads don't have detours; so the question formulation is omitting the truth -- indeed if you aren't truthful in your formulation, any puzzle would be unbeatable.
But in this case there is no missing information from the formulation -- it is truthful and complete. He even puts quotes on the labels, which indicates they are not necessarily true (but they are necessarily as written). A 'Gotcha' in this case could be for example Q:"The two labels are "..." and "..."" A:"There is no solution, because we decided to use another label!". The solution of the puzzle shown is the correct way to perform inference assuming what the statement says is true (and in real life if this happens -- that is, the inquirer is sincere -- if you chose either box you may fail).
Except he claimed it was a "logic puzzle," and he does not meet the requirements of a logic puzzle as the term is used. If you're allowing misuse of terms, then anything is possible, and my solution is as ridiculous as his, given the claim.
In fact, it's formally provable his solution makes the problem not a logic puzzle. Logic puzzles, by definition in the genre, require statements to have truth values, hence the word "logic."
If labels are either true or false, as is always the case in style of problems labeled "logic puzzles" unless the statement claims a random outcome, then his puzzle leads to the red solution.
To see the logical outcome of his "solution", by putting the treasure in the in the green box, the label "the treasure is in this box" on the green box is true.
Then the label "exactly one of the labels is true" on the red box has indeterminate truth value, because it cannot take the value true (since then both it and the green label are true), and it cannot take the value false (because then exactly one of the labels is true, contradicting the red label).
Thus, under his solution, the "logic puzzle" is not a logic puzzle. Statements cannot have indeterminate truth value by taking a final answer and working backwards and cancelling any statements that become indeterminate.
He even finished with "16 people correctly said that the treasure was in the green box. This has to be counted as a lucky guess, unacceptable as a solution to a logic puzzle."
So he claims any "correct" answer to his non-logic puzzle had to be a lucky guess? Well, he got that right - the green solution is certainly unacceptable as a solution to a true logic puzzle.
As I said, I do believe that if the formulation is abusive anything is possible. My quip is that logic puzzles shouldn't necessarily follow classical logic -- instead it should roughly be understood as "assume the statement is sincere and infer from that". Perhaps that's not the convention though.
Here's a system of logic where the puzzle has a solution:
If your logic system doesn't accept an inconsistent statement when they may occur, I tend to blame the logic system rather than reject reality. Here's a paradox illustrating my point:
> So he claims any "correct" answer to his non-logic puzzle had to be a lucky guess?
Not at all. He said the statement "the treasure was in the green box" is correct; but that is not the solution to the puzzle. The solution is "There is not enough information to determine the answer", which was explicitly provided as a possibility.
It's more of a bait-and-switch.
Here is a famous logic puzzle, often called a Knights and Knaves style puzzle [2]: Knights always tell the truth, Knaves always lie.
"John and Bill are standing at a fork in the road. John is standing in front of the left road, and Bill is standing in front of the right road. One of them is a knight and the other a knave, but you don't know which. You also know that one road leads to Death, and the other leads to Freedom. By asking one yes–no question, can you determine the road to Freedom?"
Here's a "solution" in the style of the author: "There is no solution, because the road to 'Freedom' is under construction, and the detour leads through 'Death.' Hahaha! Gotcha!"
It's a non-solution, and not a logic puzzle any more.
[1] https://en.wikipedia.org/wiki/Logic_puzzle [2] https://en.wikipedia.org/wiki/Knights_and_Knaves