1) Endings by rule / category (FIDE Laws)

At the top level a game ends in two results: win for one side or draw. The ways to reach those results are:

A. Decisive results (win–loss)

1. Checkmate (win for the side delivering mate).
2. Resignation (a player resigns; opponent wins).
3. Loss on time (a player’s time expires and the opponent has mating material; otherwise it’s a draw—see below).
4. Forfeit / default / illegal action (tournament/arbiter decision; e.g., failing to appear, illegal move in some time controls, etc.).
   This is not a “board-state” ending; it’s administrative, but it is a distinct rules-based termination in practice.

B. Drawn results

1. Stalemate (side to move has no legal move and is not in check).
2. Dead position / insufficient mating material (FIDE calls it a dead position: no sequence of legal moves can lead to checkmate, e.g., K vs K, K+N vs K, K+B vs K, etc.).
3. Draw by agreement (players agree to a draw).
4. Threefold repetition (a player claims a draw when the same position occurs three times with same side to move and same rights).
5. Fifty-move rule (a player claims a draw after 50 moves by each side with no pawn move and no capture).
   (There is also the 75-move rule: the arbiter must declare a draw automatically at 75 moves without pawn move/capture, with a small exception if checkmate occurs.)
6. Fivefold repetition (automatic draw; arbiter must declare).
7. Time expires but opponent cannot mate (time-loss is not awarded; the game is drawn).
8. Mutual flag fall (both players’ flags fall—handled by competition rules; commonly a draw unless one flag clearly fell first).

So depending on how you count (pure Laws-of-Chess vs. including tournament/arbiter outcomes), you get:

• Core “chess rules” terminations: checkmate; stalemate; dead position; draw by agreement; claim by repetition; claim by 50-move; time-loss; time-expiry draw. Plus the automatic 5-fold/75-move variants.
• Practical/tournament terminations add: forfeit/default/arbiter decisions.

A reasonable “distinct types” list people use in practice is ~8–12, depending on whether you:

• separate claimable vs automatic repetition/move-count draws, and
• include administrative forfeits separately.

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2) How many unique final positions are possible?

This is much harder, because “final position” depends on:

• what counts as a position (do you include side-to-move, castling rights, en-passant rights? For repetition rules you must),
• which termination type you mean (e.g., “final board position” after resignation is not constrained to be mate/stalemate/dead),
• legality constraints (reachable from the initial position by legal moves).

Useful known facts / bounds

A. Total number of legal chess positions (not just final)

A famous result by John Tromp gives the number of legal positions (including side to move, castling rights, and en-passant where applicable) as:

• ≈ 4.8 × 10^44 legal positions

This is widely cited and comes from careful enumeration with legality constraints (e.g., both kings present, not both in check, pawn placement constraints, etc.).

This number is for all legal positions, not just terminals.

B. Terminal positions are a subset, but still enormous

A “terminal position” (in the strict board-state sense) would usually mean one of:

• checkmate positions
• stalemate positions
• dead positions (insufficient mating material / no possible mate)

Those are all subsets of the ~10^44 legal-position universe. People have computed exact counts for endgame tablebases with limited pieces (e.g., all 7-piece positions), including which are mate/stalemate, but for full chess (up to 32 pieces) I’m not aware of a published exact count of all checkmates/stalemates/dead positions.

What is known:

• For small numbers of pieces, exact terminal counts can be derived from tablebases.
• For full chess, you can give upper bounds (≤ 4.8×10^44, obviously) and crude lower bounds (there are astronomically many distinct mate/stalemate constructions), but an exact global count is not something that’s been “closed-form” computed in the literature the way Tromp’s overall legal-position count has.

Why it’s complex to compute “all final positions”

1. Legality/reachability constraints: Many piece placements are illegal (e.g., impossible pawn structures, illegal checks, promoted pieces constraints).
2. State is more than the board: castling rights and en-passant rights affect legality and repetition; counting positions properly requires tracking them.
3. Terminal-by-rule vs terminal-by-choice:
   • After resignation or draw agreement, any legal position could be “final”, so the count becomes essentially “all legal positions that can occur in a game” (a different question).
4. Dead positions are subtle: “insufficient material” in casual talk is simpler than FIDE’s “dead position” definition (no possible mate by any legal sequence). Some positions with material can still be dead, and some “insufficient material” heuristics miss edge cases.

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If you clarify one detail, I can be more precise

For part (2), do you mean the count of terminal board states under compulsory rules (mate/stalemate/dead/flag-fall-with-no-mate), or do you also include resignation / agreed draw endings (where essentially any reachable position could be final)?

If you want, I can also give:

• counts/estimates for mate vs stalemate in N-piece tablebases (e.g., 6- or 7-piece), and/or
• a tighter discussion of “reachable positions” vs “legal positions” (those are different sets).