3 ideas
Loyd promoted a version of the fifteen-block sliding puzzle with the 14 and 15 reversed, offered a large cash prize for restoring the order, and claimed to have invented the puzzle. The prize could never be claimed. Each slide swaps the blank with one tile, and bringing the blank back to its home corner takes an even number of slides, so a single swap of two tiles, which is an odd rearrangement, can never be reached.
In the 'Get Off the Earth' paradox, rotating a disc turns 13 drawn warriors into 12, and no single figure seems to disappear. The rotation cuts each figure into slices and reassembles them so that each of the 12 new figures gets a slightly larger share. A loss spread thinly across many parts becomes invisible, because the eye counts whole units rather than tracking the total amount.
Many recreational puzzles are hard not because the math is hard but because the wording leads the solver to import an assumption the text never states. Examples include a hidden boundary, an implied order, or treating a story's sequence of events as its accounting. The fix is to separate what is actually given from what the narrative suggests, and then compute only from the givens.