Wednesday, September 29, 2004

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!!

3 comments:

Dwight Devaughn said...

Keep Blogging!

Anonymous said...

Hey Jasmine

surely it does seem impossible that anyone can just happen to solve the cube, but the maths you did just refers to one person trying to do so. If you include, say, a million people or more trying to solve the cube the odds start to look different.
which of course doesn't mean that they are high or anything, but a slight possibility might just exist...

asher said...
This comment has been removed by the author.