43,252,003,274,489,856,000 combinations

There's a discussion happening in the cube group at the moment about how ridiculous it is when non-cubers claim to have solved the cube by making random turns. Let me illustrate with some numbers.

Consider the following:

• A standard Rubik's Cube has 43,252,003,274,489,856,000 different possible configurations, only one of which is the solved state.
• There are 31,536,000 seconds in one year (60 x 60 x 24 x 365).
• If you made one turn every second for 10,000,000,000 years, you would only see 315,360,000,000,000,000 of the combinations (about 0.73% of the total number of combinations).

And here's something Stefan posted which I think is another great illustration of what we're talking about.

Original question:

"With over 43 quintillion different combinations, does anyone know the actual odds of solving it by accident? Say, 1000 random turns... can that be figured out?"

Stefan's reply:

"I had some fun learning how to use Java's BigDecimal class, here's the result (number of turns vs. probability to accidentally solve). I used 1000 digits precision.

10^0 : 0.0000000000000000000231203163
10^1 : 0.0000000000000000002312031638
10^2 : 0.0000000000000000023120316385
10^3 : 0.0000000000000000231203163852
10^4 : 0.0000000000000002312031638520
10^5 : 0.0000000000000023120316385202
10^6 : 0.0000000000000231203163852017
10^7 : 0.0000000000002312031638519936
10^8 : 0.0000000000023120316385175310
10^9 : 0.0000000000231203163849347630
10^10 : 0.0000000002312031638252929242
10^11 : 0.0000000023120316358474586107
10^12 : 0.0000000231203161179275247580
10^13 : 0.0000002312031371245709479730
10^14 : 0.0000023120289657771148207966
10^15 : 0.0000231200491127469735292260
10^16 : 0.0002311764384602375115896654
10^17 : 0.0023093609520051823138659937
10^18 : 0.0228550898430060057068061681
10^19 : 0.2064217767522700322187135016
10^20 : 0.9009402067081038970925219248
10^21 : 0.9999999999090140835069998078
10^22 : 0.9999999999999999999999999999

Isn't it funny how the numbers look very similar for a long time? And here are the reciprocal values:

10^0 : 43252003274489856000.000000000
10^1 : 4325200327448985600.4500000000
10^2 : 432520032744898560.49500000000
10^3 : 43252003274489856.499500000000
10^4 : 4325200327448986.0999500000000
10^5 : 432520032744899.05999500000000
10^6 : 43252003274490.355999500000001
10^7 : 4325200327449.4855999500000192
10^8 : 432520032745.39855999500019266
10^9 : 43252003274.989855999501926693
10^10 : 4325200327.9489855999692669303
10^11 : 432520033.24489856018766930321
10^12 : 43252003.774489857926193032100
10^13 : 4325200.8274490048668803210016
10^14 : 432520.53274509122929820999981
10^15 : 43252.503276416549031583004599
10^16 : 4325.7003467159159037864998179
10^17 : 433.02022541418460481598214744
10^18 : 43.753929950356976451685197428
10^19 : 4.8444501143894109135381508565
10^20 : 1.1099515734277697062526475446
10^21 : 1.0000000000909859165012786291
10^22 : 1.0000000000000000000000000000

So assuming that guy twisted for a full day at one twist per second, i.e. he came across 24*60*60=86400 states, the chance that he truely solved it is less than 1 out of 432520032744899."

Can you see why we speedcubers think people who claim to have solved the cube by making random turns are not quite telling the truth!!