3x3 Speedcubing: The Ultimate Limits Forecast

State Space 4.3×10¹⁹

3x3 has 43,252,003,274,489,856,000 ≈ 4.3 × 10¹⁹ reachable states:

|G| = (8! · 3^7) · (12! · 2^11) / 2 = 4.3252 × 10^19

  8 个角的位置: 8! = 40,320
  7 个角的朝向 (第 8 个被约束): 3^7 = 2,187
  12 个棱的位置: 12! = 479,001,600
  11 个棱的朝向 (第 12 个被约束): 2^11 = 2,048
  角棱位置奇偶性必须同号 → 除 2

素因子分解: |G| = 2^27 · 3^14 · 5^3 · 7^2 · 11

• Corner orientation sum ≡ 0 mod 3. Each has 3 twist states; the 8th is forced.
• Edge orientation sum ≡ 0 mod 2. F/B quarter turns flip 4 edges; 12th is forced.
• Corner-edge parity coupling. Every face quarter turn is a 4-cycle on both corner and edge ring (both odd) — they stay locked. This is the /2.

Universe scale. Universe is ~4.36 × 10¹⁷ s old. Enumerate one state per second: 1.37 trillion years ≈ 100× the age of the universe.

Info-theoretic. Identifying a state takes log₂(4.3 × 10¹⁹) ≈ 65.22 bits ≈ 9 bytes.