# New estimate for 20f*: 300,000,000

Of these 250 random cosets, 232 had absolutely no distance-20 positions. The 18 cosets with distance-20 positions had a total of 33 distance-20 positions; 10 had a single distance-20 position, 5 had two distance-20 positions, and the remaining distance-20 positions were contained in cosets containing 4, 5, and 6 such positions, respectively.

With these new results, I now estimate that there are about 300,000,000 distance-20 positions overall. Luckily, we know which cosets they are concentrated in, and I am currently exploring those cosets out to completion to try to find as many of the distance-20 positions as I can.

In the table below, you'll see pretty good agreement between the exact values at the 250 coset values at distances 12 and 13. At lesser distances, the accuracy falls off because the "hit" sample size is much smaller.

The overall agreement between the 100K positions solved by Kociemba and the 250 cosets I have solved is very good.

The estimate for the count of 20f* has fallen markedly from my initial experiment with only 25 cosets. Because of this it is fairly likely that the current estimate of about 300 million is also not extremely accurate, but I believe it is in the correct order of magnitude.

d exact percentage 250 cosets percentage 100K percen -- --------------- ----------- ------------- ------------ ----- ------ 0 1 2.31203e-18 1 18 4.16166e-17 2 243 5.61824e-16 3 3240 7.49098e-15 4 43239 9.99699e-14 5 574908 1.32921e-12 6 7618438 1.76141e-11 7 100803036 2.3306e-10 8 1332343288 3.08042e-09 30 6.15119e-10 9 17596479795 4.06836e-08 990 2.02989e-08 10 232248063316 5.36965e-07 18603 3.81435e-07 11 3063288809012 7.08242e-06 291034 5.96735e-06 12 40374425656248 9.33469e-05 4191486 8.59421e-05 13 531653418284628 0.0012292 57843281 0.00118602 14 - 779147743 0.0159756 18 0.018 15 - 10312034775 0.211438 197 0.197 16 - 131362659765 2.69345 2710 2.71 17 - 1315341261754 26.9697 26673 26.673 18 - 3261117757529 66.8658 67099 67.099 19 - 158131992975 3.24233 3303 3.303 20 - 35 7.17639e-10