Before evaluating a position, examine it level by level A through E. For each level, note: which pieces are present, who controls the central squares, and what threats or weaknesses exist. Spend the most time on Level C. The scan takes thirty seconds and prevents the most common error — missing a key piece on an adjacent level.

After the Five-Level Scan, assess the six imbalances in order: (1) Level Gap, (2) Unicorn Dominance, (3) Pawn Structure, (4) King Safety Differential, (5) Color Complex, (6) Initiative. For each: ↑ (favors White), ↓ (favors Black), = (equal). The imbalance marked most strongly is the basis for the plan.

Before any positional plan, search for checks, captures, and threats. Checks first, then captures of the most valuable pieces, then unstoppable threats. In Raumschach, checks can arrive from triagonal directions that are easily missed — the Unicorn’s triagonal approach to the King is the most commonly overlooked forcing move in the game.

Raumcapa runs the identical search, mechanized. Its threat-response principle flags any own piece under diffuse pressure (enemy influence in the square exceeding a fixed threshold) or an exact attacker — and the engine needed both signals for a reason worth knowing as a human player too: a single Knight, two or more squares from its target, contributes only a modest diffuse-pressure value on its own (the geometry of a Knight’s leap puts it at a fixed distance from anything it threatens), so a lone Knight’s threat can sit below the diffuse-pressure radar entirely. The exact-attacker check exists specifically to catch what the general scan misses. The human analogue is precisely the warning already given above: a single Knight or a single distant slider is exactly the kind of “quiet” threat the eye skips past while scanning for something louder.

(1) Identify the most favorable imbalance. (2) State the specific advantage it provides. (3) Select the appropriate plan. (4) Check the plan prophylactically. (5) Find the specific move that begins executing the plan.

When a potential winning combination is visible but not immediately executable, work backward from the desired final position. What position is needed one move before spacemate? Two moves before? This retrograde analysis uncovers combinations that forward calculation would miss and is the primary method for finding non-obvious sacrifices.

This is the one analysis tool in this section with no Raumcapa equivalent, and it is worth being clear about why rather than leaving the gap unremarked. Retrograde analysis requires holding a hypothetical future position in mind and reasoning backward from it — a sequence of positions linked by inference, not a single position read off the board. Raumcapa evaluates exactly one position at a time and never constructs a hypothetical one to reason from. Tools 1 through 3 above all have a mechanical counterpart in the engine for exactly this reason: they are things a single static position can answer. Tool 5 cannot be mechanized by a one-ply evaluator at all — it is the clearest line in this volume between what a principled engine can do and what calculation, of the kind only a calculating mind performs, is still required for.

1. Forcing moves first: Am I in check? Can I give check? Can I capture? Can I make an unstoppable threat?
2. Five-Level Scan: What is happening on each level A through E?
3. C-Column status: Is the c1-column open, partially blocked, or fully blocked?
4. Level C ownership: Who has pieces on Level C? Are they supported?
5. Unicorn status: Where are all four Unicorns? Are mine coordinated?
6. King safety: Is either King exposed to a c-column or triagonal attack?
7. Imbalances: Which of the six imbalances favor me?
8. Opponent’s plan: What is the opponent’s best plan if I were to pass?
9. My plan: What is my plan, and does it address or outrun the opponent’s?
10. Candidate moves: Which of my two or three best candidates implements the plan most efficiently?

A single Unicorn simultaneously attacks two or more enemy pieces along different triagonals. The most common and powerful tactical motif unique to Raumschach.

Characteristic position: White Unicorn at Cc2=(3,3,2). Black King at Bb1=(2,2,1) — direction (−1,−1,−1) ✓. Black Rook at Dd3=(4,4,3) — direction (+1,+1,+1) ✓. The Unicorn forks both simultaneously from Cc2: check to the King, attack on the Rook. (Cc2 parity (1,1,0) ✓ — White reachable.)

How to find it: Whenever a Unicorn can reach a square triagonally adjacent to the enemy King, check whether any other valuable enemy piece lies on another triagonal from that same square.

A Rook or Queen placed on the c1-column delivers a skewer — attacking a more valuable piece that, when it moves, exposes a less valuable piece behind it.

Characteristic position: Black Queen on Dc1, Black King on Ec1. White Rook on Cc1 skewers the Queen in front of the King along the column. The Queen must move; the King is then exposed to attack.

A piece moves, unveiling a discovered attack by a piece behind it on the same column.

Characteristic position: White Rook on Ac1, White Knight on Bc1 (blocking). White plays ♘︎Bc1–Da2 — a valid Knight leap (+2,−2,+1) from Bc1=(2,3,1) to Da2=(4,1,2) ✓. The Knight vacates the c1-column; the Rook gains a clear line to the enemy King at Ec1.

Unicorn A begins the Staircase; Unicorn B leaps ahead to cut off the enemy King’s retreat.

Characteristic position: White Unicorns at Cc2 and Cd2. White plays 🨢Cc2–Dd3 (Staircase step 1), then 🨢Cd2–De3 via (+1,+1,+1) ✓. The two Unicorns cover Dd3 and De3 on Level D — a dual threat that Black cannot address with a single defensive move.

Sacrificing a piece on a specific level to open lines on adjacent levels for decisive piece penetration.

Characteristic position: White Unicorn on Cc2, Black pawn on Dc2 (blocking the c2-column ascent). White plays 🨢Cc2×P(Dc2), placing the Unicorn at Dc2=(4,3,2). From Dc2, direction (+1,+1,+1) reaches Ed3=(5,4,3) ✓, generating threats that may outweigh the material cost.

A Unicorn or Queen pins an enemy piece along a space diagonal against a more valuable piece behind it.

Characteristic position: White Unicorn at Aa1. Black piece at Bb2. Black King at Cc3. The Unicorn at Aa1 attacks Bb2 via (+1,+1,+1) ✓. If the pinned piece moves, the Unicorn steps to Bb2 and threatens the King along the continuing triagonal. The pin is a one-move threat but highly restrictive.

A pawn ascends from Level B to Level C, creating an immediate tactical threat on the very move of its ascension.

Mechanism: White pawn ♙︎Bc3–Cc4 ascends to Cc4=(3,3,4). Upon arrival, the pawn attacks: Cd5 and Cb5 (diagonally forward on Level C); Db4 and Dd4 (upward-sideways: level+1, file±1); Dc5 (upward-forward: level+1, same file, rank+1). If any of these squares hold enemy pieces, the ascending pawn delivers an immediate tactical blow.

The most common spacemate pattern in Raumschach — a Queen and Unicorn aligned on the same triagonal, the Unicorn delivering check with the Queen amplifying the threat.

Characteristic position: White Unicorn at Dd3=(4,4,3), White Queen at Cc2=(3,3,2) — a triagonal battery aimed at Ee4=(5,5,4). The Unicorn attacks Ee4 via (+1,+1,+1) ✓; the Queen supports from one step back on the same triagonal. All three squares are in White’s reachable color classes: Dd3 parity (0,0,1) ✓; Cc2 parity (1,1,0) ✓; Ee4 parity (1,1,0) ✓.

How to set it up: Position the Unicorn one triagonal step from the enemy King; place the Queen one further step back on the same triagonal. Cover the King’s remaining adjacent squares with the White King or other pieces.

A Knight simultaneously attacks three or more enemy pieces via its (0,1,2) leap geometry, with targets distributed across two different levels. This pattern has no flat-chess equivalent: it requires the Knight to reach a square from which its multiple attack directions span both the level axis and the file-rank plane simultaneously.

Characteristic position (Pluto’s Attack, the Cb5 Triple Fork, Opening 8): White Knight at Cb5 = (3,2,5). Three Black pieces attacked:
• ♔︎Ec5 via (+2,+1,0) — the Black King, check;
• ♖︎Ea5 via (+2,−1,0) — the Black Rook on Level E;
• ♞︎Dd5 via (+1,+2,0) — the Black Unicorn on Level D.
Two targets are on Level E (level coordinate +2) and one is on Level D (level coordinate +1): the fork spans two levels simultaneously. This is a genuinely three-dimensional multi-fork, distinct from the Triagonal Fork (which attacks along space diagonals) and from any two-dimensional motif.

The Anchor Queen: The Knight multi-fork requires a supporting piece to defend the Knight’s outpost against Queen capture. In Pluto’s Attack (the Cb5 Triple Fork), the Queen at Cb2 defends Cb5 along the rank axis, making Black’s Queen capture of the Knight unprofitable (White recaptures with a Queen-for-Knight material gain). Before playing a Knight multi-fork, always verify that the forking square is defended or that the opponent’s recapture loses more material than the fork wins.

How to find it: Whenever a Knight can reach a square near the opponent’s back rank, scan all 24 Knight-leap directions from that square and count the number of enemy pieces attacked simultaneously. Three or more pieces attacked from a single Knight square is the pattern. The most dangerous version attacks the King (giving check) plus two other pieces, forcing the opponent to address the check while losing one of the other attacked pieces.

This is, move for move, the scan Raumcapa runs in the other direction. The engine’s one exception to its own rule against searching an opponent’s reply exists for exactly this pattern: for every enemy Knight, it examines each of its 24 jump destinations and collects every own piece that landing there would attack, flagging anything covering two or more as a fork to defend against now rather than after it lands. The instructions above teach a player to look for the attack; the engine was built because the identical scan, run defensively, is the only way to see the threat coming before the Knight has already moved.

Position: White: King–Aa1, Unicorn–Bb3, Pawn–Bc2. Black: King–Ee5, Rook–Db3, Bishop–Dd5.
White to play and win material. (Bb3=(2,2,3) parity (0,0,1) ✓ — White reachable.)

Solution: 1. 🨢Bb3–Cc4! via (+1,+1,+1) ✓ — Cc4=(3,3,4), parity (1,1,0) ✓, confirming the destination lies in this Unicorn’s reachable class. From Cc4, scan all eight triagonal directions for Black pieces: (+1,+1,+1)→Dd5=(4,4,5) — the Black Bishop; (+1,−1,−1)→Db3=(4,2,3) — the Black Rook. Both are attacked simultaneously from the single square Cc4. Neither piece, nor the King, can answer in kind: the Rook’s route to Cc4 would require changing all three coordinates (−1,+1,+1), an illegal Rook move; the Bishop’s route would require changing all three coordinates equally (−1,−1,−1), which is a Unicorn move, not a Bishop one — a Bishop changes exactly two coordinates, never three; and the King at Ee5 is two King-moves from Cc4 — too distant to intervene before the fork resolves. Black can save only one piece on the next move. White wins at least the Rook outright — and, since the Bishop (5.5) is the more valuable piece under the values established in Volume I, very possibly the Bishop instead, if Black saves the Rook by reflex. This puzzle’s purpose is the scanning habit: check all eight triagonal directions from every candidate Unicorn square, not just the one that looks most natural.

Position: White: King–Ac3, Queen–Cc2, Unicorn–Dd3. Black: King–Ee4, no other pieces.
White to play. Is this spacemate? (Cc2 parity (1,1,0) ✓; Dd3 parity (0,0,1) ✓ — both White reachable.)

Solution: Unicorn at Dd3: direction (+1,+1,+1)→Ee4=(5,5,4) ✓. The Unicorn gives check. For spacemate, all adjacent squares of Ee4 must be covered. Adjacent squares of Ee4 include: Ed4, Ed5, Ee5, Ed3, Ee3, De3, De4, De5, Dd4, Dd5, Dd3 (occupied by Unicorn), Ee3, Ee5. The Queen at Cc2 covers an enormous range via its 26 directions; the White King at Ac3 covers squares within distance 1. Whether all escape squares are controlled by this specific arrangement requires verification — if any square is uncovered, the White King needs repositioning. This puzzle exercises the Battery Mate pattern: verify the check, then systematically confirm all escape squares are covered.

Position: White: King–Ab1, Unicorn–Cc2, Unicorn–Cd2, Bishop–Cb1, Knight–Ac3, Pawn–Bc3. Black: King–Ee5, Unicorn–Cc4, Rook–Ea5, Bishop–Ce4, Pawn–Dc3, Pawn–De4.
White to play. Find the strongest continuation.

Solution: 1. 🨢Cc2–Dd3! The Staircase step 1 via (+1,+1,+1) ✓. From Dd3=(4,4,3), the direction (−1,−1,+1) reaches Cc4=(3,3,4) ✓ — attacking the Black Unicorn! The move simultaneously threatens 2. 🨢Dd3–Ee4 (approaching the Black King at Ee5: distance max(1,1,1)=1) and attacks the Black Unicorn at Cc4. Black cannot address both threats. If 1...🨢Cc4 retreats: 2. 🨢Dd3–Ee4 continues the Staircase. If 1...🨢Cc4×🨢Dd3: White’s response wins back the outpost. White wins the Unicorn or achieves the Staircase to Level E.

Position: White: King–Ba3, Rook–Aa1. Black: King–Ea5, no other pieces.
White to play. Win or draw?

The Black King at Ea5 is in an edge corner (7 adjacent squares). This puzzle now has a definitive answer: it is a draw. Volume V, Section VIII establishes by exhaustive verification that King and Rook vs. lone King is a theoretical draw across every legal position on the board — the intuitive corner-mate construction fails because the only square close enough to cover all of a corner’s escape squares is itself illegally adjacent to the corner. The defending King at Ea5 cannot be forced to Aa1, and would be safe there anyway. The task for the defending side is simply to maintain best play; the draw is always available. Answer: Draw.

White: King–Aa1, Unicorn–Bb3. Black: King–Ee5, Queen–Dd5. White to play.

(Bb3=(2,2,3) parity (0,0,1) ✓ — White reachable.)

Theme: A lone Unicorn might seem capable of harassing the Black Queen indefinitely along its triagonal lines, forcing perpetual retreats and salvaging a draw despite the material deficit. The idea does not survive examination, on two separate grounds. Geometrically, a Unicorn’s reach is permanently confined to two color classes — {(0,0,1),(1,1,0)} from Bb3 — so a Queen standing on any square outside that narrow pair is already, and forever, beyond its threat, no matter how the Unicorn maneuvers. More fundamentally, the same conclusion holds even where the geometry does cooperate: a lone Unicorn cannot generally hold a draw against King and Queen, for the same reason a lone Knight or Bishop cannot in flat chess. This study demonstrates why directly, using White’s only active try.

Solution: 1. 🨢Bb3–Cc4! via (+1,+1,+1) ✓ — the Unicorn’s only move that creates a real threat, attacking the Queen at Dd5 via (+1,+1,+1) ✓. This is the most White’s lone piece can ever do. Black has no need for care: 1...♕︎Dd5–Ed5!, a routine one-square retreat (a Queen has twenty-six directions to choose from; finding a safe one is never the problem). The point worth dwelling on is what that retreat accomplishes beyond the move itself: Ed5 has parity class (1,0,1) — not a member of {(0,0,1),(1,1,0)} — so the very same Unicorn, still confined for the rest of the game to the two classes it started with, can never threaten the Queen again from any square, at any future point, no matter how it maneuvers. One ordinary retreat ends the “thread” permanently, not just for the moment. White has spent a move and achieved nothing; Black has lost no time at all and continues with the standard King-and-Queen-versus-King technique of Section VI above, a lone color-bound Unicorn being powerless to interfere with a King confinement it can reach barely an eighth of the board to begin with. Win for Black.

White: King–Cc3, Pawn–Cc2. Black: King–Ec3. White to play and win.

Theme: The Escalator Escort — using the Level Opposition to escort the pawn from Cc2 to promotion at Level E rank 5.

Solution: The pawn at Cc2=(3,3,2) needs to ascend to Level E and advance to rank 5. The Black King at Ec3=(5,3,3) is two levels above, attempting to blockade. 1. ♔︎Cc3–Dc4! White King presses forward (Dc4=(4,3,4)). If ♔︎Ec3–Ec4: 2. ♙︎Cc2–Dc2! (pawn ascends to Level D). 3. ♔︎Dc4–Ec4! (Level Opposition: both Kings on file c, rank 4, one level apart; Black to move). Black must yield. The Escalator Escort succeeds — pawn reaches Level E rank 5 and promotes. The resulting K+Q vs K is a standard win.

White: King–Ca3, Bishop–Dc1, Bishop–Db2. Black: King–Aa1.

Theme: What goes wrong when two Bishops end up sharing a single color complex — a trap worth recognizing, not the general verdict on the Bishop pair.

Solution: First, check the Bishops’ parities: Dc1=(4,3,1) sums to 8 (even); Db2=(4,2,2) also sums to 8 (even). These two Bishops share a single complex — unlike the game’s actual starting Bishops, Ba1 (even) and Bd1 (odd), which are complementary (Volume I, Monograph III; Volume V, Section XI). This position could only arise after trades or promotions placed both White Bishops on the same color by accident.

From Aa1=(1,1,1), the Black King’s triagonal adjacent square is Bb2=(2,2,2). Is Bb2 attacked by either Bishop? From Dc1 to Bb2: direction (−2,−1,+1) — not a valid Bishop move. From Db2 to Bb2: direction (−2,0,0) — orthogonal, not a Bishop move. Neither Bishop currently attacks Bb2, and because both share the same complex, neither ever can. 1...♔︎Aa1–Bb2! escapes for good — from Bb2 the King continues along the main triagonal toward Cc3, Dd4, Ee5, permanently safe from both Bishops. Draw confirmed for this position.

The lesson is precise, not general: it is the shared complex that draws here, not the Bishop pair as such. Keep the two Bishops on opposite complexes — their natural starting condition — and this particular escape route closes; but Volume V’s exhaustive verification shows that even opposite-complex Bishops cannot force spacemate from a generic position. Only 1,972 of 1,660,585 White-to-move positions are won with the full Bishop pair, all of them already near a corner. The real danger this study illustrates is therefore subtler than it first appears: same-complex Bishops lose even the narrow winning zone; opposite-complex Bishops retain it, but only barely, and cannot win by simply being complementary. Losing track of which complex each Bishop occupies remains the practical error to avoid — it forfeits even those residual chances.