Unsolvable means impossible to solve or figure out. When a problem is unsolvable, no amount of thinking, calculating, or experimenting will lead to an answer because no solution exists.

In mathematics, certain problems have been proven unsolvable. For centuries, mathematicians tried to square the circle (construct a square with the exact same area as a circle using only a compass and straightedge), until they finally proved it was unsolvable. Some puzzles, like certain chess positions, can be unsolvable: there's simply no way to win from that position, no matter how brilliant your strategy.

Outside of math, people sometimes call problems unsolvable when they just seem impossibly difficult. A teacher might say a conflict between two students feels unsolvable, meaning she can't figure out how to fix it, even though a solution probably exists. In this looser sense, unsolvable means “extremely difficult” rather than truly impossible.

Sometimes, people discover that a problem really has no solution at all, and then they may choose to focus their effort on other challenges they actually can solve.