Monte Carlo Simulation: Probability of 6-Move 3 Edge Cycle Case
We wrote a Monte Carlo simulation program which generates a random 3 edge cycle, and make N random moves, and identify whether at certain point, there is a 6 move 3 edge cycle case, e.g. M' U2 M U2. The result is following
| Number of Moves | Probability of 6-Move 3 Edge Cycle Case | Probability of 6 or 8-Move 3 Edge Cycle Case (with setup) |
|---|---|---|
| 1 | 1.20% | 4.00% |
| 2 | 1.92% | 5.16% |
| 3 | 2.36% | 6.91% |
| 4 | 2.90% | 8.10% |
| 5 | 3.39% | 9.69% |
| 6 | 3.97% | 9.98% |
| 7 | 3.98% | 11.14% |
| 8 | 4.38% | 12.38% |
| 9 | 4.68% | 13.52% |
| 10 | 5.24% | 14.88% |
| 11 | 5.60% | 15.92% |
| 12 | 5.81% | 16.88% |
| 13 | 6.45% | 18.28% |
| 14 | 6.96% | 18.51% |
| 15 | 7.58% | 19.92% |
| 16 | 7.30% | 21.09% |
| 17 | 7.81% | 22.48% |
| 18 | 8.43% | 23.22% |
| 19 | 8.97% | 23.36% |
| 20 | 9.34% | 24.67% |
| 21 | 9.61% | 25.92% |
| 22 | 9.70% | 26.51% |
| 23 | 10.47% | 27.34% |
| 24 | 10.56% | 28.83% |
| 25 | 10.88% | 29.20% |
| 26 | 12.13% | 29.18% |
| 27 | 12.42% | 30.77% |
| 28 | 12.15% | 31.90% |
| 29 | 12.64% | 33.00% |
In practice, you will likely be doing an insertion from 20-30 moves, so the probability of 6 mover is around 12% and does not increase so much even if you have more moves; the probability of 8 mover is 25 to 33%.
This knowledge might be helpful for you to set expectations when you find a promising skeleton, so you are aware of the expected outcome after an insertion, and decide whether to move forward with an edge insertion. The 6 move edge insertion is relatively easy to identify, compared to corner commutator though.