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logic problem
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The Scorchio Thief was tired of the small change he was getting from the cashiers at the Neopian National Bank, so he broke in one night to raid the vault. When he got there, he came to a combination lock on the vault, with the dial numbers going from 0 to 59. Unfortunately, he wasn't sure whether there were three or four numbers in the combination, or even which direction to turn the wheel! If it takes him 15 seconds to try a single combination, how many days will it take him to to try every possible combination? Please round to the nearest day.
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That's actually a Permutation lock, in a combination series (like the National Lottery) the order doesn't matter, but in Permutations (like what is commonly referred to a Combination lock) order does matter.
Anyway
Possible Combinations
With 3 numbers:
60*59*59 combinations, x 2 for allowing to start either way.
(Including zero there are 60 numbers to choose from, but two consecutive numbers couldn't be the same, otherwise the dial wouldn't need turing)
=417 720 combinations
With 4 numbers:
60*59*59*59*2
=24 645 480 combinations
=25 063 200 total possible
Taking 15 seconds each gives 375 million seconds, which is 4351.25 days (around 12 years)
There are 60 ways of choosing the first number
The second number must be different from the first number, but could be reached either way, so there are (59*2) ways of choosing the second number
The third number must be different from the second number (but could be the same as the first) but could be reached either way, so there are (59*2) ways of choosing the third number
Similarly for the fourth number
So there are
60*(59*2)*(59*2)*(59*2) ways of choosing four numbers
that�s 98581920 ways
There are
60*(59*2)*(59*2) ways of choosing three numbers
that�s 835440 ways
Total
99417360 at 15 seconds per go
24854340 minutes
414239 hours
17259.9 days
47 years
He'd need to sleep and attend to other bodily functions. He'll probably also spend a bit of time dodging the guards -- and perhaps trying to read the number over their shoulders when they open the safe to put even more money in.
Then, what if he forgets where he's got to?
On the up-side, with practice he'd probably get each try well below 15 seconds -- though why does a 3-number set take the same time as a 4-number one.
However, would he not be able to try each 3-number permutation on the way to the 4-number ones? In which case you could ignore the time needed for the whole 3-number set.
After a while, he might start wondering if there were actually five numbers, or if the number had been changed that time back in 1972 when he was on holiday...
Or the lock might wear out altogether.
Come on, this isn't really a very practical question, now is it...?
I bought a car once which had a key-number for the radio. The previous owner had lost the number, so I sat down one evening to work through the permutations.
Lucky for me, the number began with 13, so it only took about 15 minutes. Also lucky there was no limit to the number of tries, nor a delay between each one. Less lucky, I couldn't listen to the radio while I did it.
So, subsidiary real-life practical question: how many digits in my radio key?
(My current car has a key number for the ignition. If you get it wrong three times, you have to wait half an hour before trying again...)
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