The problem seems to advance by successive movements, and the natural temptation is to explore sequences. But the answer comes before starting to try combinations.
Ten pieces and the black impossible
Riddle statement
You start with 10 white pieces on the table.
On each move you must choose exactly two pieces and turn them over: white goes to black and black goes to white.
Is it possible to ever get into a situation with exactly one black piece?
Show solution
Solution
Answer: No, it is impossible.
Each move turns over exactly two pieces. Depending on the chosen combination, the number of black pieces changes by $-2$, $0$ or $+2$ — always an even amount. The parity of the number of black pieces is, therefore, an invariant.
You start with 0 black pieces (even).
Reaching 1 black piece would require an odd number.
That contradicts the invariant: no sequence of moves, no matter how long, can cross that parity barrier.