[Retros] Variables

[Joost de Heer]

In the latest Problemist, there's an article about variables and AP 
validation of certain variables moves.

Variable: A piece of known colour but unknown type, which can play as 
any orthodox piece (or fairy piece already present in the diagram). All 
possible legal substitutions of Variables for pieces are to be 
considered. In the play, only moves consistent with legal substitutions 
up to that point are legal. If a variable moves, only its departure and 
arrival square are considered. After each move, certain substitutions 
may no longer be possible, being inconsistent with the play so far. 
Captures, checks, mate and stalemate are only effective if they are 
consistent with all remaining substitutions.

Examples:

A. Christian Poisson
Rex Multiplex 1993
8/8/8/8/8/6K1/1V6/7k (2+1)
Variable b2
a) #1
b) =1

a) 1. Vb1! The variable can only be a queen or rook, and either one 
gives mate. 1. Va1? isn't mate, as the variable can be either a queen or 
a bishop.
b) 1. Vd4! The variable can only be a queen or bishop, and either one 
gives mate. Similar to a), Vb6? isn't stalemate, as the variable can be 
either a queen or a rook.

In diagram A, 1. Vb2-b6 Kh1-g1 is a legal sequence. White's move proves 
that the variable is a queen or a rook, black's move proves it's a rook.

B. Tadashi Wakashima
8/8/8/8/2vR4/8/1V6/k1K5 (3+2)

1. Vb2-b4 Vc4-b3 2. Rd4-a4#

With white's second move, he proves that both variables were pawns, and 
that black's first move was an ep capture.

C. Stephen Emmerson
8/6kV/8/8/8/8/8/4K2R (3+1)

1. Vh7-h8 proves that on h7 there was a pawn. Promotion of a variable 
pawn is to a variable officer, so to determine the exact nature of the 
promotion, further moves are necessary.
1. Ve1-g1 Kg7-g6. This proves that the variable on e1 was a king, by 
eliminating the possibilities of it being a queen or rook, and therefore 
white's first move was castling, Vh1 was a rook and is now on f1.

Originals:

D. Geoff Foster
4vVVV/VV2Vvvv/4V3/3vVVV1/3V4/8/8/8 (11+5)
+---+---+---+---+---+---+---+---+

|   | . |   | . |*V | V | V | V |

+---+---+---+---+---+---+---+---+

| V | V | . |   | V |*V |*V |*V |

+---+---+---+---+---+---+---+---+

|   | . |   | . | V | . |   | . |

+---+---+---+---+---+---+---+---+

| . |   | . |*V | V | V | V |   |

+---+---+---+---+---+---+---+---+

|   | . |   | V |   | . |   | . |

+---+---+---+---+---+---+---+---+

| . |   | . |   | . |   | . |   |

+---+---+---+---+---+---+---+---+

|   | . |   | . |   | . |   | . |

+---+---+---+---+---+---+---+---+

| . |   | . |   | . |   | . |   |

+---+---+---+---+---+---+---+---+
h==9, Variables

1. Vh7-h5 Vg5-h6 2. Vg7-g5 Vf5-g6 3. Vf7-f5 Ve5-f6 4. Ve8-b5[=bBQ, 
d5=bK] Ve7-h7[=wRQ] 5. Kd5-c6 Vb7-g7[=wR] 6. 
Vb5-h5!![=bQ,f6=wP,g6=wP,h6=wP] Va7-f7[=wRQ] 7. Qh5-e5 Ve6-e7[=wP] 8. 
Kc6-d7[vf8~S] Vd4xe5[=wPB] 9. Kd7-e8[Vf8=B,Vf7=R] Ve5-e6[=wP]==

Three e.p. captures on move 1-3 are validated on move 6. Note that not 
all variables are known (Vh7=wRQ, Vg8!=wB, Vh8!=wB), but all 
substitutions consistent with the previous play lead to stalemate. The 
beginposition is: 4qBVV/RR2Vppp/4P3/3kPPP1/3P4/8/8/8

E. Ian Shanahan
8/v7/8/1vV5/v4v2/6v1/6v1/4V2V (3+6)
+---+---+---+---+---+---+---+---+

|   | . |   | . |   | . |   | . |

+---+---+---+---+---+---+---+---+

|*V |   | . |   | . |   | . |   |

+---+---+---+---+---+---+---+---+

|   | . |   | . |   | . |   | . |

+---+---+---+---+---+---+---+---+

| . |*V | V |   | . |   | . |   |

+---+---+---+---+---+---+---+---+

|*V | . |   | . |   |*V |   | . |

+---+---+---+---+---+---+---+---+

| . |   | . |   | . |   |*V |   |

+---+---+---+---+---+---+---+---+

|   | . |   | . |   | . |*V | . |

+---+---+---+---+---+---+---+---+

| . |   | . |   | V |   | . | V |

+---+---+---+---+---+---+---+---+
ser-=7, Variables

1. Vc5-b6 [b6!=wRS, a7!=bK] 2. Vb6xa7 3. Va7-a8[[c5]=wKQP, b5!=bK, 
a8!=wP] 4. Va8xg2 [g2=wQB;[c5]=wQP] 5. Ve1-g1 [g1=wQRK] 6. Vf1xf4! [5. 
OO!, g1=wK, [h1]=wR, a4!=bK, g3=bK, g2=wB => [c5]=wP, 1. cb6ep!] 7. Rf4xa4=
Monochromatic Valladao task, ideal stalemate.

F. Mark Ridley
8/8/8/8/1Vv5/8/V1R5/4V1v1 (2+4)
+---+---+---+---+---+---+---+---+

|   | . |   | . |   | . |   | . |

+---+---+---+---+---+---+---+---+

| . |   | . |   | . |   | . |   |

+---+---+---+---+---+---+---+---+

|   | . |   | . |   | . |   | . |

+---+---+---+---+---+---+---+---+

| . |   | . |   | . |   | . |   |

+---+---+---+---+---+---+---+---+

|   | V |*V | . |   | . |   | . |

+---+---+---+---+---+---+---+---+

| . |   | . |   | . |   | . |   |

+---+---+---+---+---+---+---+---+

| V | . | R | . |   | . |   | . |

+---+---+---+---+---+---+---+---+

|[.]|   | . |   | V |   |*V |   |

+---+---+---+---+---+---+---+---+
ser-h#3, Variables, magic square a1

1. Vc4-b3 [b3~bRS, a2~wK] 2. Vb3xa2 [a2~bK, g1=bK] 3. a2-a1=wV[[c4]=bQP, 
b4~wK, e1=wK] Ke1-c1 => OOO# => d1=R => [c4]=P => 1. cb3ep => b4=wP]
Valladao, all three aspects involve the same variable.

A Fairyable is a Variable that can only be substituted by a fairy piece 
already present in the diagram.

G. Stephen Emmerson
2V5/6vG/5Pl1/1Kp5/1p3kl1/1L5G/7P/6G1 (8+6)
+---+---+---+---+---+---+---+---+

|   | . | V | . |   | . |   | . |

+---+---+---+---+---+---+---+---+

| . |   | . |   | . |   |*V | G |

+---+---+---+---+---+---+---+---+

|   | . |   | . |   | P |*L | . |

+---+---+---+---+---+---+---+---+

| . | K |*P |   | . |   | . |   |

+---+---+---+---+---+---+---+---+

|   |*P |   | . |   |*K |*L | . |

+---+---+---+---+---+---+---+---+

| . | L | . |   | . |   | . | G |

+---+---+---+---+---+---+---+---+

|   | . |   | . |   | . |   | P |

+---+---+---+---+---+---+---+---+

| . |   | . |   | . |   | G |   |

+---+---+---+---+---+---+---+---+
h#2; fairyables c8, g7; grasshopper g1, h3, h7; locust b3, g4, g6
b) Gg1->h8

a) 1. Vg7-e5 Gh7-f5 [e5=bL, -wPf6] 2. Lg6xf5-e4 Vc8-c4 (c4=L? self-check 
from e5, since c5 is captured => c4=G => #)
b) 1. Vg7-g5[=bG => +) Gh3-f5 2. Lg4xf5-e6+ Vc8-c4 (c4=G?? => c4=L => #).
B1 sets up the proof of the wV on W2 by the attack on the wK, proving 
the attack on the bK.