Mobility in Reserve

White plays. Find his only best first shot and the final differential that he can guarantee against any optimal defense.

This page contains the solution.

The answer

White must play H1: differential guaranteed +16.

The reasoning

Start by establishing the legal blows by looking, in each of the eight directions, for an unempty adverse suite ended with a pawn of your color. A blow in a corner is not automatically the best.
Each sheet is evaluated by the number of pawns of the camp at the beginning minus those of the other camp. At a node where this camp plays, one keeps the maximum; at an opposing node, the minimum. A pass changes the line without consuming any box; two camps without legal blow complete the part.
The exact values of the first shots, after the best defence: H1: +16; A2: -20; A3: -16; G7: +0; A8: -12; C8: -22; G8: +2. The maximum is unique: H1.
After H1, all the opposing legal answers must be processed. Here is an optimal continuation of the two camps (N = Black, B = White); the score is always brought back to the camp which was to play at the start:
Adverse response H4: B A8, N G7, B G8, N D8, B A3, N A2, B C8. End: 24 black, 40 white; initial differential +16.
Opposite response G7: B H4, N D8, B A8, N pass, B C8, N pass, B G8, N pass, B A3, N A2. Finish: 22 black, 42 white; initial differential +20.
Adverse response C8: B H4, N D8, B A8, N G7, B G8, N pass, B A3, N pass, B A2. Finish: 15 black, 49 white; initial differential +34.
Adverse response D8: B H4, N G7, B A8, N pass, B C8, N pass, B G8, N pass, B A3, N A2. End: 22 black, 42 white; initial differential +20.
The opponent retains the answer giving the smallest of these differentials: 16. The other answers therefore do not refute the first blow; they obtain an equal or better result for you.
Complete main line: B H1, N H4, B A8, N G7, B G8, N D8, B A3, N A2, B C8. Final check: 24 black and 40 white. The blank boxes are not assigned to anyone.
Technique: Minimal retrograde calculation, mobility and pass management, then comparison of filling orders. Note for each branch available blows rather than assuming an uninterrupted alternation. The table of values is an exhaustive search-assisted proof; continuations alone would not prove that the omitted defences are inferior.

Common mistakes and false leads

Count only the pawns returned by the next shot.
Forget a diagonal or a mandatory pass.
Consolidate a main line with evidence against all defences.

How we verified this answer

Legal part generation from the standard initial position; exhaustive minima stored to the end, without truncated depth or leaf heuristic. All the values of the first strokes are checked.

{
  "empty": 9,
  "legal_first_moves": 7,
  "first_move_scores": {
    "H1": 16,
    "A2": -20,
    "A3": -16,
    "G7": 0,
    "A8": -12,
    "C8": -22,
    "G8": 2
  },
  "states": 16489,
  "history": [
    "D3",
    "E3",
    "F3",
    "C3",
    "C5",
    "B5",
    "D6",
    "C7",
    "F5",
    "F4",
    "G5",
    "F2",
    "D2",
    "D1",
    "B6",
    "A7",
    "B8",
    "G6",
    "B3",
    "E6",
    "F7",
    "H5",
    "B4",
    "G3",
    "C1",
    "B1",
    "B7",
    "E2",
    "F1",
    "C6",
    "C4",
    "C2",
    "D7",
    "A5",
    "F6",
    "G1",
    "H3",
    "G2",
    "E1",
    "G4",
    "A1",
    "A6",
    "H6",
    "E7",
    "H2",
    "A4",
    "H7",
    "E8",
    "B2",
    "H8",
    "F8"
  ],
  "player": -1,
  "best_score": 16,
  "principal_variation": [
    "B H1",
    "N H4",
    "B A8",
    "N G7",
    "B G8",
    "N D8",
    "B A3",
    "N A2",
    "B C8"
  ]
}

Several first moves are legal; an estimate by the immediate gain does not replace the calculation until the end. The resolution requires a minimum tree on several floors, with verification of the turns and mobility. Editorial evaluation based on the required techniques; not calibrated by a human panel.

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