Queen vs Rook

The defender's strategy is simple and maddening: keep the rook adjacent to the king (or one knight's-move away where it can't be forked), decline every trade, and check whenever your king steps forward carelessly. Separated from its king, the rook falls to a queen fork within a few checks; every defender knows it, so the two pieces move as one huddled unit toward the safest edge.

Your assets: the queen out-ranges the rook absolutely, your king is allowed to join the attack (the defender's king can never help its rook), and zugzwang; the huddle has very few safe shuffles, and you can run it out of them.

Phase one: herd. Use queen checks and king steps to push the defending pair to the board's edge. Centralize your queen, walk your king up under its cover: no hurry, no unforced checks.

Phase two: zugzwang the huddle. The classic finale is the Philidor queen-vs-rook position: defending king on the edge, rook beside it, your king a knight's move from theirs, and it's their move. Every rook move now steps out of the king's protection into a forking radius; every king move abandons the rook or walks into mate threats. If the same position arrives with you to move, triangulate with the queen (the queen loses a tempo effortlessly; a rook-king huddle cannot) to hand the move back. Phase three: collect. Chase the fleeing rook with checks until a fork wins it; then it's the basic queen mate.

The defender's best practical try is the third-rank defense: king on the back rank, rook planted on its third rank, building a fence your king cannot cross without being harassed by checks from the side. It looks like a fortress. It isn't: the queen infiltrates behind the fence with a precise sequence of quiet moves and checks, attacks the rook from behind, and forces it off the rank, after which the huddle-and-zugzwang plan resumes.

This is the ending where practicing against a tablebase pays fastest: perfect defense makes every human shortcut fail, and the 31-move budget punishes wandering. Once you've beaten the machine, the human version feels like a formality.