Why did solving chess endings with one more piece take ten more years?

Because of combinatorial explosion: the rapid growth in a problem's complexity as its inputs grow. By 2005, every chess ending with six pieces or fewer had been solved, with the result of each position under perfect play recorded in a 'tablebase'. Adding one more piece added so much combinatorial complexity that the seven-piece tablebase took ten more years to complete, and an eight-piece one is considered intractable. That is also why chess, with only 64 squares and 32 pieces, is still not a solved game.

The pattern shows up everywhere once you look. A Sudoku grid is a special kind of Latin square, and as the grid gets bigger, a combinatorial explosion limits which Sudokus can be constructed, analyzed and solved. Even counting how many Latin squares exist as their size grows is a textbook example of the same explosion.

The same explosion appears with something as simple as true-or-false switches. One Boolean variable has two possible states, but each additional variable doubles the count: two variables give four states, three give eight, and n variables give two to the power of n. That doubling compounds brutally fast.

It even shows up in something as human as who needs to talk to whom. Two departments need one communication channel; add a third, and you need three channels; a fourth needs six; a fifth needs ten; a sixth needs fifteen. People often call this growth 'exponential', but it's actually a slower, if still painful, polynomial pattern, following the same formula as counting how many pairs you can make from a group.