These three puzzles, about the ball and bat, the widget-making machines and the lily pads, are the components of Shane Frederick's 'Cognitive Reflection Test'.
It was given to students at Princeton; half of them saw it normally on a computer monitor, the other half saw it as greyed text on a grey background. The idea was that the latter group would experience 'cognitive strain', meaning that they had to work to get the correct answer.
However, 90% of those who saw the text normally made at least one mistake, but only 35% of those who saw the hard-to-read version. Further, in those who who see the questions normally, around one-third get all the questions wrong. You might well wonder at these statistics—the questions didn't fool people here. The 'Cognitive Reflection Test' is widely used in psychological profiling.
The idea is that the subject is pre-programmed to get an obvious, intuitive answer; the correct answer needs some brain-work. The intuitive thinking is the 'fast' system or System 1; the brain-work is the 'slow' system, or System 2.
The point is that there are many activities where we don't need to think about what to do, we don't waste processing power or time, for the 'slow' system takes a while to get the correct answer.
Psychologists have tested this in other ways; they put people in an unknown room; the subject sees a door, or at least recognises a door, and is told to open it and leave the room. However, the door knob is on the same side of the door as the hinges, so the door doesn't open when the knob is turned. It takes some effort to work out why the door won't open.
You can see the intuitive or 'fast' system at work all the time; the problem is when some people, such as (gullible) journalists or politicians leap to the obvious answer. If a persuasive person or organisation said that, say. Ballyjamesduff was the Irish centre of a vast trafficking movement, such people would take this as being correct without thinking about it. (One American town/city was said to be a trafficking hub because it had a canal.)
The other puzzles are also examples of where the intuitive answer is wrong; for the rope round the earth, the idea is that the added length is so tiny by comparison to the earth's diameter, that the rope 'must' be only a few millimetres away from the surface; the Monty Hall problem gives the 'obvious' answer that it doesn't matter whether you stay with the first choice, or move to the second. It is, however, much harder to get to the correct answer.
It's not that 'fast' or intuitive thinking is wrong; if you were in the jungle and heard a rustle in the background, you might wonder if it's a tiger looking to eat you or something else. But you should stick with the first thought—that's it's a hungry tiger—and flee. Similarly, if you have a 'gut-feeling' that there's something wrong with a client or escort, you are right to trust your intuition.