Quandaries and Queries


A friend asked me to solve this for their 12th grade math homework

Franklin's friend had taken an item from him, and put it in his family's safe, when franklin went to retrieve it he came to a combination lock on the safe, 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.



Hi Adam,

I am not sure if combinations on safes work like combinations on padlocks but I am going to assume that they do. My recollection of padlocks is that you turn to the first number, reverse direction and turn back, past the first number to the second number, and then reverse again and turn to the third number. I seems to remember also that the second number can not be the same as the first. If this is not the way you see the combination lock working then you might have to change some of my arithmetic.

Under my rules there are 60 possibilities for the first number, and once you have decided on a first number there are 59 choices for a second number, and then 60 choice for a third number. Thus for a three digit combination lock there are

60 59 60 = 212 400

combinations if you know which direction to start with. Since you don't, you need to double this number to get

2 212 400 = 424 800 combinations.

It takes 15 seconds to try a combination and hence 4 combinations can be attempted in a minute. Thus you can try 4 60 = 240 combinations in an hour or 24 240 = 5 760 combinations in a day. Since there are 442 800, three digit combinations, to try them all would take

 442 800/5 760 = 73.75 days.

Now you do the four digit combinations.



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