They made a 15 puzzle where instead of numbers the tiles had letters on them, and the goal was to make a particular sentence. That sentence had 13 letters that appeared once and one letter that appeared twice.
That letter that appeared twice was the first letter of the sentence, and its other appearance was at or near the end. Let's call the two tiles with the same letter T1 and T2, where T1 is the tile that is the tile that is first.
They would show someone the solved puzzle, then scramble it, and challenge the person to solve it. When they were scrambling it they made sure to move T2 to the first position and T1 to somewhere near the end.
Nearly everyone who tried to solve it would see the right letter at the first position and never move it, because most people try to solve left to right and/or top to bottom.
On the 15 puzzle exactly half of the possible permutations of the 15 tiles are reachable from a given starting position. Exchanging two tiles in a reachable permutation produces an unreachable permutation.
That means that the right sentence but with T1 and T2 swapped is not reachable and so people who left T2 in front were doomed to failure.
I know it happens on the 3x3x4 cuboid - each of the 3x4 faces has two centers that appear identical, but if you have exactly two such unsolved pieces, it's unsolvable because it's impossible to exchange exactly two of them - what you need to do is cycle 3 of them where 2 appear identical. I remember this happening on some other shapes and sizes as well.
(Pic for reference: https://www.grubiks.com/images/puzzles/30/small.png . This puzzle exists as do many other NxMxH cuboids. Rectangular but non-square faces can only make 180º turns.)
Did you mean flip an edge, or 120°?
If you got to the point where you can routinely solve cubes, i.e. you know your algorithms or you developed a good intuition, then unsolvable cubes should be pretty obvious. Some algorithms are like: if you are in this position, do this, if you are in that position, do that, other positions are illegal. If you are in an illegal position, then you know there is a problem with your cube.
The Rubik's cube is a really hard puzzle unless you know some theory, and the existence of unsolvable cubes is a rather well known fact, meaning that there shouldn't be that many people who are able to solve the cube and don't know about unsolvable cubes.
Or maybe that's just me. I learned on a terrible cube that had a tendency to dismantle itself, so knowing that it had to be put back properly was a requirement for a successful solve.