White 138 vs Black 40
White starts from setup 138 (NRQNBBKR), Black from setup 40 (NBNQBRKR). In the encyclopedia since 2026-10-03: neither side starts better.
|cp| ≤ 5 at 400M · 800M · 1.6B · 4B nodes, two independent engines agree
The two setups
White: setup #138 N R Q N B B K R
King on the g-file, rooks on b, h, queen on the c-file; no bishop starts in a corner; the knights start away from the centre. The king already stands on a castled square, so one of the castlings moves only the rook. In ordinary symmetric play setup 138 gives White +0.45 (Morozov Balance Index, 400M nodes).
Black: setup #40 N B N Q B R K R
King on the g-file, rooks on f, h, queen on the d-file; no bishop starts in a corner; the knights start away from the centre. The king already stands on a castled square, so one of the castlings moves only the rook. In ordinary symmetric play setup 40 gives White +0.34 (Morozov Balance Index, 400M nodes).
Why the pair is balanced
At the deepest rungs (400M, 800M, 1.6B nodes) the evaluation is exactly 0.00 each time, reaching search depth 62.
Admission rule: |cp| ≤ 5 at four budgets up to 4 billion nodes, no one-sided drift along the ladder, and confirmation by two independent engines — Berserk and Lc0 — each below its own median on symmetric positions.
The shallow rungs wandered between -21 and +36 cp before settling — a reminder that quick engine checks overstate imbalances, and why the ladder goes deep.
For comparison, in symmetric play setup 138 gives White +0.45 and setup 40 gives White +0.34: with both sides on the same rank the first move is worth something. In this pair Black's own setup absorbs that edge completely.
White is not forced into a single path: the best first move is h4, and 3 first moves are within 10 cp of it.
Along the engine's main line (25 moves) the evaluation stayed within ±13 cp the whole way: the balance holds as the pieces come out, not only on move one.
Engine self-play agrees: 10 games, White won 0, drawn 10, Black won 0 (White scored 50.0%).
Verification ladder
Nine independent searches of the starting position, each with a larger node budget. Evaluation in centipawns from White's point of view.
| Stage | Nodes | Depth | Eval, cp | Main line |
|---|---|---|---|---|
| S0 | 100 000 | 15 | +36 | 1. Nb3 f5 2. d4 c6 3. h4 Nd6 4. e3 Ne4 |
| S1 | 500 000 | 17 | -3 | 1. Nb3 d5 2. c4 c6 3. h4 h5 4. d4 Nd6 |
| S2 | 2 000 000 | 26 | -21 | 1. h4 h5 2. d4 d5 3. c4 c6 4. Nb3 Nd6 |
| S3 | 8 000 000 | 32 | -9 | 1. h4 h5 2. c4 c6 3. Nb3 d5 4. d4 Nab6 |
| S4 | 32 000 000 | 37 | +4 | 1. h4 h5 2. c4 c6 3. Nb3 d5 4. d4 Nd6 |
| S5 | 128 000 000 | 46 | 0 | 1. d4 c6 2. Nb3 d5 3. c4 h5 4. h4 Nd6 |
| S6 | 400 000 000 | 55 | 0 | 1. d4 c6 2. Nb3 d5 3. c4 h5 4. h4 Nd6 |
| S7 | 800 000 000 | 60 | 0 | 1. d4 c6 2. Nb3 d5 3. c4 h5 4. h4 Nd6 |
| S8 | 1 600 000 000 | 62 | 0 | 1. d4 c6 2. Nb3 d5 3. c4 h5 4. h4 Nd6 |
Deep verification
The search ladder reaches 1.6 billion nodes; admission needs a 4-billion rung and two further engines on top of that. Beyond even that, every pair was re-checked at budgets up to ten times larger and in thousands of self-play games. This section shows what each check returned for this pair.
| Nodes | Eval, cp |
|---|---|
| 4 000 000 000 | not checked |
| 8 000 000 000 | not checked |
| 16 000 000 000 | not checked |
This pair held exactly 0.00 up to 1 600 000 000 nodes.
Independent engines, run on the same position:
- Berserk 14: +2
- Lc0 0.32.1: +9
Independent confirmations: 2 of 3. An engine counts as confirming when its reading of the pair is closer to equality than its own reading of a typical symmetric Chess960 position (+10 for Berserk, +18 for Lc0). Where only one engine confirms, the pair is balanced according to that engine alone, and this page says so.
Own weight of each setup, measured against the classical array: White +38, Black +52, difference -14. A balanced pair needs White's setup to be weaker by about the price of the first move.
Extended self-play: 10 games, 0 White wins, 10 draws, 0 Black wins, White scored 50.0%.
Best first moves
The 3 best first moves for White, searched together at 400 000 000 nodes.
| # | Eval, cp | Depth | Main line |
|---|---|---|---|
| 1 | +3 | 45 | 1. h4 e5 2. c4 h6 3. b4 c6 4. d4 exd4 5. Nb3 d5 |
| 2 | -1 | 44 | 1. c4 c6 2. h4 h5 3. Nb3 d5 4. d4 Nd6 5. cxd5 cxd5 |
| 3 | -4 | 44 | 1. Nb3 h5 2. h4 c6 3. c4 d5 4. d4 Nd6 5. cxd5 cxd5 |
Main-line play-out
25 moves, 5 000 000 nodes per half-move. Largest |evaluation| 13 cp, mean 2.5 cp, final 0 cp.
| # | White | cp | Black | cp |
|---|---|---|---|---|
| 1 | h4 | -13 | h5 | -10 |
| 2 | c4 | -8 | c6 | -9 |
| 3 | d4 | -8 | d5 | -4 |
| 4 | Nb3 | 0 | Nd6 | +4 |
| 5 | cxd5 | +2 | cxd5 | +4 |
| 6 | Nc3 | 0 | Nc7 | -11 |
| 7 | e3 | 0 | f6 | -2 |
| 8 | Qc2 | 0 | Bf7 | 0 |
| 9 | Rd1 | 0 | b6 | 0 |
| 10 | Bd3 | -5 | Qd7 | 0 |
| 11 | f3 | 0 | Rc8 | 0 |
| 12 | Qe2 | 0 | Nf5 | 0 |
| 13 | Bf2 | +12 | O-O | +13 |
| 14 | O-O | +12 | g6 | +2 |
| 15 | e4 | 0 | dxe4 | 0 |
| 16 | fxe4 | 0 | Nh6 | 0 |
| 17 | d5 | -6 | Ng4 | 0 |
| 18 | Nd4 | 0 | e6 | 0 |
| 19 | Nc6 | 0 | exd5 | 0 |
| 20 | exd5 | 0 | Rfe8 | 0 |
| 21 | Qf3 | 0 | Nxd5 | 0 |
| 22 | Nxb8 | 0 | Qc7 | 0 |
| 23 | Bg3 | 0 | Qc5+ | 0 |
| 24 | Bf2 | 0 | Qc7 | 0 |
| 25 | Bg3 | 0 | Qc5+ | 0 |
| 26 | Bf2 | 0 |
Engine self-play
10 games at 500 000 nodes per move: White won 0, drawn 10, Black won 0 — White scored 50.0%. Average length 37.1 moves.
Sample game 1 (1/2-1/2)
1. c4 c6 2. h4 d5 3. h5 f5 4. Nb3 Rf6 5. e3 Nc7 6. f4 Rh6 7. Be2 dxc4 8. Qxc4+ Nd5 9. Qc2 O-O 10. Nc3 Ncb6 11. Rh3 Qd7 12. Nc5 Qc8 13. Nd3 Nxc3 14. bxc3 c5 15. c4 Nd7 16. Bh4 Bxh5 17. Bxh5 Rxh5 18. Bxe7 Rxh3 19. Bxf8 Rg3 20. Be7 b6 21. Ne1 Qe8 22. Kf2 Rg6 23. Bh4 Rd6 24. Rd1 Qf7 25. Nf3 h6 26. g3 Re6 27. Qa4 Nf8 28. Qb5 Rd6 29. d4 cxd4 30. exd4 Rd7 31. Re1 Bd6 32. Re2 Ne6 33. Kg2 Nf8 34. Kf2 Ne6 35. Kg2 Nf8
Sample game 2 (1/2-1/2)
1. Nb3 d5 2. h4 c6 3. c4 h5 4. d4 Nd6 5. cxd5 cxd5 6. Nc3 Nb6 7. Nc5 Bc6 8. b4 Bd7 9. f3 Bc8 10. a4 Nf5 11. e3 e5 12. dxe5 Bxe5 13. a5 Nc4 14. Bf2 b6 15. axb6 Nxb6 16. Nb5 Ng3 17. Nxa7 Nxh1 18. Kxh1 Rh6 19. b5 Bd7 20. f4 Bc7 21. Ra1 Nc4 22. Nxd7 Qxd7 23. Nc6 Bb6 24. Ra6 Qc7 25. Bxc4 dxc4 26. Qxc4 Re8 27. Qb3 Rf6 28. Kh2 g6 29. Qd3 Rd6 30. Qe2 Rde6 31. Qc4 Kg7 32. Qd3 Re4 33. Qd5 R4e6 34. Qd3 Re4 35. Qd5 R8e6 36. Nd4 Bxd4 37. Rxe6 Rxe6 38. Qxd4+ f6 39. Be1 Qb7 40. Qc5 Qb6 41. Bb4 Qxc5 42. Bxc5 Re8 43. b6 Rb8 44. e4 f5 45. e5 Kf7 46. Kg3 Ke6 47. Kf2 Kd5 48. Be3 Kc6 49. Ke2
Play it in the CH960 app
Open this page on a phone with the CH960 app installed, or find pair 138 / 40 in the app's encyclopedia, then choose how to play.
- Local game — two players on one device, with a clock.
- With a mentor — a clearly labelled training bot with hints, at your level.
- Challenge a friend — an online game by link; both players get the same pair, colours swap on rematch.
Play this pair against the computer, right here Play a friend
FEN (Shredder, castling rights by rook files): nbnqbrkr/pppppppp/8/8/8/8/PPPPPPPP/NRQNBBKR w BHfh - 0 1
Mirror twin (the same position mirrored): White 821 vs Black 925.