Lucky-Scramble Forecast: Cumulative Probability of Hitting the Luckiest Scramble

3x3 has 4.3×10^19 states; only 262 (= 1+18+243) are solvable in ≤2 moves. A single random scramble has only ~6×10^-18 chance of landing there. But WCA accumulates many scrambles per year (incl. backups); cumulative-hit probability is P = 1 − (1 − p)^N, monotonically rising with N. Drag the year → see N(Y) → see cumulative probability per depth threshold → see the expected luckiest result.

Year

2026

Cumulative 3x3 scrambles
N(Y) = 6.61M (incl. backups)

Left half (0–40%) is linear 2003–2100; right half is log up to 10^15 yr.

3x3: Cumulative Hit Probability

Through 2026, WCA has accumulated 6.61M 3x3 scrambles (incl. backups). Cumulative probability of at least one d ≤ K hit across these N is P = 1 − (1 − p)^N. Smaller K → smaller p in the 4.3×10^19 state space → need exponentially larger N.

Depth K (moves)	# states at d ≤ K	Single-scramble p	P(d≤K hit) @ N=6.61M	N₅₀ (scrambles)	Year reached	Time at d (s, TPS 17)
d ≤ 2	262	6.1×10^-18	4.00×10^-11	1.14×10^17	2.0×10^6	0.27 s
d ≤ 5	621,649	1.44×10^-14	9.50×10^-8	4.82×10^13	41,996	0.44 s
d ≤ 8	1.44B	3.33×10^-11	2.20×10^-4	20.80B	2746	0.62 s
d ≤ 10	251.29B	5.81×10^-9	3.77%	119.31M	2056	0.74 s
d ≤ 12	4.37×10^13	1.01×10^-6	99.9%	686,214	2017	0.86 s
d ≤ 15	9.89×10^16	0.229%	~100%	303	2004	1.03 s

How to read: e.g. d ≤ 2 row — single-scramble prob 6×10^-18 (262 of 4.3×10^19). Need N₅₀ ≈ 1.2×10^17 to have 50% chance of one such hit; that's ~10^14 yr away at WCA's 7.5M scrambles/year cap. That is why "3x3 2-move lucky solve" is not a near-future event but an asymptotic limit. Today the expected luckiest scramble is d ≈ 11.5 with P(d≤12) ≈ 99.9%.

3x3 · 2026 luck forecast

0.83 s

E[min] = 11.56 moves / 17 TPS + 0.15 s

Cumulative scrambles

6.61M

2003–2026 · incl. backups

Expected min depth

11.56 moves

WCA accepts ≥2 moves

TPS 14.6 (real cuber)

0.94 s

TPS at Wang sustained level

Cumulative Hit Probability vs Year

Each line corresponds to one K threshold. X-axis: log year (2003 → 10^15 yr), Y-axis: P(hit ≤ K). Watch when each line crosses from 0 to 1.

P(d ≤ 2)P(d ≤ 5)P(d ≤ 8)P(d ≤ 10)P(d ≤ 12)P(d ≤ 15)

Expected Luckiest Time vs Year (Cross-Event)

Log year vs log seconds. Each line = E[min depth] / TPS_ceil + setup_s per event. Asymptote = k_min_wca / TPS_ceil + setup (fastest physically possible single).

3x3x32x2x24x4x45x5x5PyraminxSkewb

Per-Event Forecast at Selected Year

★ = exact depth distribution; ◐ = partial (exact low / est. high); ~ = approximate. Time = E[min] / TPS_ceil + setup_s.

3x3x3333◐

Cumulative scrambles

6.61M

E[min] depth

11.56

Expected luckiest

0.83 s

11.6m / 17 TPS + 0.15s

2x2x2222★

Cumulative scrambles

5.62M

E[min] depth

1.00

Expected luckiest

0.16 s

1.0m / 16 TPS + 0.1s

4x4x4444~

Cumulative scrambles

4.63M

E[min] depth

30.55

Expected luckiest

3.35 s

30.6m / 12 TPS + 0.8s

depth distribution approximated

5x5x5555~

Cumulative scrambles

3.64M

E[min] depth

46.64

Expected luckiest

5.44 s

46.6m / 11 TPS + 1.2s

depth distribution approximated

6x6x6666~

Cumulative scrambles

1.65M

E[min] depth

69.91

Expected luckiest

9.27 s

69.9m / 9 TPS + 1.5s

depth distribution approximated

7x7x7777~

Cumulative scrambles

1.32M

E[min] depth

95.00

Expected luckiest

13.87 s

95.0m / 8 TPS + 2s

depth distribution approximated

3x3 One-Handed333oh◐

Cumulative scrambles

2.64M

E[min] depth

11.87

Expected luckiest

1.39 s

11.9m / 10 TPS + 0.2s

Clockclock~

Cumulative scrambles

1.65M

E[min] depth

6.91

Expected luckiest

0.68 s

6.9m / 13 TPS + 0.15s

depth distribution approximated

Megaminxminx~

Cumulative scrambles

1.19M

E[min] depth

45.04

Expected luckiest

6.00 s

45.0m / 10 TPS + 1.5s

depth distribution approximated

Pyraminxpyram★

Cumulative scrambles

3.64M

E[min] depth

1.00

Expected luckiest

0.18 s

1.0m / 13 TPS + 0.1s

Skewbskewb★

Cumulative scrambles

2.64M

E[min] depth

1.00

Expected luckiest

0.18 s

1.0m / 13 TPS + 0.1s

Square-1sq1~

Cumulative scrambles

1.85M

E[min] depth

11.84

Expected luckiest

1.48 s

11.8m / 10 TPS + 0.3s

depth distribution approximated

Methodology

State space. 3x3 = 4.3252 × 10^19 states; 2x2 = 3,674,160; Pyraminx = 933,120; Skewb = 3,149,280. Their full depth distributions are known.
Distribution sources. 3x3: cube20.org / Rokicki 2010 (exact d=0..15, est. d=16..20). 2x2: Korf / Pochmann. Pyraminx / Skewb: Jaap Scherphuis. Larger cubes etc.: peak-concentrated approximation.
Accumulated scrambles N(Y). 2003–2025 from WCA dump (≈28k comps total), 2026+ extrapolated at 5% CAGR capped at 30k comps/year. Per-comp 3x3 scrambles (incl. backups) linearly interpolated 30 → 250 (2003 → 2026). Other events by share factor (4x4 ≈ 0.70 × 3x3).
Cumulative probability. P(hit at least one d ≤ K in N draws) = 1 − (1 − p_le_K)^N, with p_le_K = ∑_{i≤K} counts[i] / |S|. Implemented via log1p(-p) for numerical stability when p is tiny.
Expected min E[min]. E[min depth] = ∑_{k=0}^{G-1} (1 − P(min ≤ k)), then clamped to WCA-acceptable minimum (3x3 ≥ 2; 2x2 / Pyraminx / Skewb ≥ 1).
Execution time. T = E[min] / TPS_ceil + setup_s. TPS_ceil = physiological ceiling (3x3 = 17, dual-hand 22 Hz drum × 50% loss), OH = 10, big cubes 8–12. setup_s = trigger + tap-off 0.10–2.00 s.
Asymptote (N → ∞). 3x3: 2 moves / 17 TPS + 0.15 s = 0.27 s. 2x2 / Pyraminx / Skewb: 1 move / TPS_ceil + setup ≈ 0.16–0.20 s. These are reachable only when luck + TPS are both maxed; the physical floor (M/TPS+R) is still ~1.5 s and bounds real-comp WRs.

Caveats

WCA scrambler is uniform sampling, but TNoodle outputs 17–25 move sequences (not filtered for short-solve states). A cumulative hit is mathematical; in practice the cuber would also have to recognize "this is a d=4 state" mid-comp and find a 4-move optimal solution (no algorithm book covers low-d states).
Recognition + switching + reaction + dual-hand coordination floor (≥ 50 ms StackMat trigger) mean the "d=2 + 17 TPS" 0.27 s is unreachable; the physical floor is ~1.5 s.
Large cubes / Megaminx / Sq1 / Clock distributions not enumerated; peak-concentrated approximation. Order-of-magnitude reliable, not single-digit.
FMC / blind not modeled (FMC time = move count; blind is memo-bound, not TPS).

Lucky-Scramble Forecast: Cumulative Probability of Hitting the Luckiest Scramble

Year

2026

Cumulative 3x3 scrambles
N(Y) = 6.61M (incl. backups)

Left half (0–40%) is linear 2003–2100; right half is log up to 10^15 yr.

3x3: Cumulative Hit Probability

Depth K (moves)	# states at d ≤ K	Single-scramble p	P(d≤K hit) @ N=6.61M	N₅₀ (scrambles)	Year reached	Time at d (s, TPS 17)
d ≤ 2	262	6.1×10^-18	4.00×10^-11	1.14×10^17	2.0×10^6	0.27 s
d ≤ 5	621,649	1.44×10^-14	9.50×10^-8	4.82×10^13	41,996	0.44 s
d ≤ 8	1.44B	3.33×10^-11	2.20×10^-4	20.80B	2746	0.62 s
d ≤ 10	251.29B	5.81×10^-9	3.77%	119.31M	2056	0.74 s
d ≤ 12	4.37×10^13	1.01×10^-6	99.9%	686,214	2017	0.86 s
d ≤ 15	9.89×10^16	0.229%	~100%	303	2004	1.03 s

3x3 · 2026 luck forecast

0.83 s

E[min] = 11.56 moves / 17 TPS + 0.15 s

Cumulative scrambles

6.61M

2003–2026 · incl. backups

Expected min depth

11.56 moves

WCA accepts ≥2 moves

TPS 14.6 (real cuber)

0.94 s

TPS at Wang sustained level

Cumulative Hit Probability vs Year

Each line corresponds to one K threshold. X-axis: log year (2003 → 10^15 yr), Y-axis: P(hit ≤ K). Watch when each line crosses from 0 to 1.

P(d ≤ 2)P(d ≤ 5)P(d ≤ 8)P(d ≤ 10)P(d ≤ 12)P(d ≤ 15)

Expected Luckiest Time vs Year (Cross-Event)

Log year vs log seconds. Each line = E[min depth] / TPS_ceil + setup_s per event. Asymptote = k_min_wca / TPS_ceil + setup (fastest physically possible single).

3x3x32x2x24x4x45x5x5PyraminxSkewb

Per-Event Forecast at Selected Year

★ = exact depth distribution; ◐ = partial (exact low / est. high); ~ = approximate. Time = E[min] / TPS_ceil + setup_s.

3x3x3333◐

Cumulative scrambles

6.61M

E[min] depth

11.56

Expected luckiest

0.83 s

11.6m / 17 TPS + 0.15s

2x2x2222★

Cumulative scrambles

5.62M

E[min] depth

1.00

Expected luckiest

0.16 s

1.0m / 16 TPS + 0.1s

4x4x4444~

Cumulative scrambles

4.63M

E[min] depth

30.55

Expected luckiest

3.35 s

30.6m / 12 TPS + 0.8s

depth distribution approximated

5x5x5555~

Cumulative scrambles

3.64M

E[min] depth

46.64

Expected luckiest

5.44 s

46.6m / 11 TPS + 1.2s

depth distribution approximated

6x6x6666~

Cumulative scrambles

1.65M

E[min] depth

69.91

Expected luckiest

9.27 s

69.9m / 9 TPS + 1.5s

depth distribution approximated

7x7x7777~

Cumulative scrambles

1.32M

E[min] depth

95.00

Expected luckiest

13.87 s

95.0m / 8 TPS + 2s

depth distribution approximated

3x3 One-Handed333oh◐

Cumulative scrambles

2.64M

E[min] depth

11.87

Expected luckiest

1.39 s

11.9m / 10 TPS + 0.2s

Clockclock~

Cumulative scrambles

1.65M

E[min] depth

6.91

Expected luckiest

0.68 s

6.9m / 13 TPS + 0.15s

depth distribution approximated

Megaminxminx~

Cumulative scrambles

1.19M

E[min] depth

45.04

Expected luckiest

6.00 s

45.0m / 10 TPS + 1.5s

depth distribution approximated

Pyraminxpyram★

Cumulative scrambles

3.64M

E[min] depth

1.00

Expected luckiest

0.18 s

1.0m / 13 TPS + 0.1s

Skewbskewb★

Cumulative scrambles

2.64M

E[min] depth

1.00

Expected luckiest

0.18 s

1.0m / 13 TPS + 0.1s

Square-1sq1~

Cumulative scrambles

1.85M

E[min] depth

11.84

Expected luckiest

1.48 s

11.8m / 10 TPS + 0.3s

depth distribution approximated

Methodology

State space. 3x3 = 4.3252 × 10^19 states; 2x2 = 3,674,160; Pyraminx = 933,120; Skewb = 3,149,280. Their full depth distributions are known.
Distribution sources. 3x3: cube20.org / Rokicki 2010 (exact d=0..15, est. d=16..20). 2x2: Korf / Pochmann. Pyraminx / Skewb: Jaap Scherphuis. Larger cubes etc.: peak-concentrated approximation.
Accumulated scrambles N(Y). 2003–2025 from WCA dump (≈28k comps total), 2026+ extrapolated at 5% CAGR capped at 30k comps/year. Per-comp 3x3 scrambles (incl. backups) linearly interpolated 30 → 250 (2003 → 2026). Other events by share factor (4x4 ≈ 0.70 × 3x3).
Cumulative probability. P(hit at least one d ≤ K in N draws) = 1 − (1 − p_le_K)^N, with p_le_K = ∑_{i≤K} counts[i] / |S|. Implemented via log1p(-p) for numerical stability when p is tiny.
Expected min E[min]. E[min depth] = ∑_{k=0}^{G-1} (1 − P(min ≤ k)), then clamped to WCA-acceptable minimum (3x3 ≥ 2; 2x2 / Pyraminx / Skewb ≥ 1).
Execution time. T = E[min] / TPS_ceil + setup_s. TPS_ceil = physiological ceiling (3x3 = 17, dual-hand 22 Hz drum × 50% loss), OH = 10, big cubes 8–12. setup_s = trigger + tap-off 0.10–2.00 s.
Asymptote (N → ∞). 3x3: 2 moves / 17 TPS + 0.15 s = 0.27 s. 2x2 / Pyraminx / Skewb: 1 move / TPS_ceil + setup ≈ 0.16–0.20 s. These are reachable only when luck + TPS are both maxed; the physical floor (M/TPS+R) is still ~1.5 s and bounds real-comp WRs.

Caveats

WCA scrambler is uniform sampling, but TNoodle outputs 17–25 move sequences (not filtered for short-solve states). A cumulative hit is mathematical; in practice the cuber would also have to recognize "this is a d=4 state" mid-comp and find a 4-move optimal solution (no algorithm book covers low-d states).
Recognition + switching + reaction + dual-hand coordination floor (≥ 50 ms StackMat trigger) mean the "d=2 + 17 TPS" 0.27 s is unreachable; the physical floor is ~1.5 s.
Large cubes / Megaminx / Sq1 / Clock distributions not enumerated; peak-concentrated approximation. Order-of-magnitude reliable, not single-digit.
FMC / blind not modeled (FMC time = move count; blind is memo-bound, not TPS).