SOLO-036 · Othello / Reversi

The Last Word

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 A8: guaranteed differential +6.

The reasoning

1. 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.
2. 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.
3. The exact values of the first shots, after the best defense: A1: -6; C1: +4; A2: -30; A8: +6; B8: -2; F8: -6. The maximum is unique: A8.
4. After A8, you have to deal with all the opposing legal answers. Here is an optimal continuation of the two camps (N = Black, B = White); the score is always brought back to the camp that was to play at the start:
5. Adverse response C1: B B8, N F8, B A1, N H1, B A2, N G1, B H2, N H4, B H5. End: 29 black, 35 white; initial differential +6.
6. Opposite response G1: B B8, N C1, B A1, N A2, B H1, N H2, B H4, N F8, B H5. End: 22 black, 42 white; initial differential +20.
7. Adverse response H2: B B8, N F8, B A2, N H4, B H5, N pass, B A1, N pass, B C1, N pass, B G1, N pass, B H1. End: 16 black, 48 white; initial differential +32.
8. Adverse response H4: B B8, N F8, B H5, N G1, B A1, N A2, B C1, N pass, B H1, N H2. End: 24 black, 40 white; initial differential +16.
9. Adverse response H5: B B8, N H4, B A1, N F8, B A2, N G1, B C1, N pass, B H1, N H2. End: 27 black, 37 white; initial differential +10.
10. Adverse response B8: B F8, N H1, B C1, N A1, B A2, N G1, B H2, N H5, B H4. End: 28 black, 36 white; initial differential +8.
11. Adverse response F8: B B8, N H5, B H4, N C1, B A1, N G1, B H1, N H2, B A2. Finish: 26 black, 38 white; initial differential +12.
12. The opponent retains the answer giving the smallest of these differentials: 6. The other answers therefore do not refute the first blow; they obtain an equal or better result for you.
13. Complete main line: B A8, N C1, B B8, N F8, B A1, N H1, B A2, N G1, B H2, N H4, B H5. Final check: 29 blacks and 35 whites. The blanks are not assigned to anyone.
14. 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.

{
  "empty": 11,
  "legal_first_moves": 6,
  "first_move_scores": {
    "A1": -6,
    "C1": 4,
    "A2": -30,
    "A8": 6,
    "B8": -2,
    "F8": -6
  },
  "states": 688850,
  "history": [
    "C4",
    "C5",
    "C6",
    "E3",
    "F5",
    "B6",
    "A6",
    "D6",
    "E6",
    "E7",
    "F7",
    "C3",
    "D3",
    "G5",
    "D7",
    "C7",
    "F6",
    "G8",
    "F2",
    "E2",
    "D8",
    "C8",
    "D2",
    "E8",
    "B3",
    "B2",
    "B4",
    "A5",
    "F3",
    "E1",
    "B1",
    "G6",
    "A4",
    "A3",
    "H7",
    "F4",
    "D1",
    "G7",
    "G4",
    "A7",
    "H8",
    "C2",
    "H6",
    "G3",
    "F1",
    "B5",
    "H3",
    "G2",
    "B7"
  ],
  "player": -1,
  "best_score": 6,
  "principal_variation": [
    "B A8",
    "N C1",
    "B B8",
    "N F8",
    "B A1",
    "N H1",
    "B A2",
    "N G1",
    "B H2",
    "N H4",
    "B H5"
  ]
}

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