#MathProblemOfTheDay: Interesting Combination Lock


I saw this combination lock at a Duane Reade.  It has only four options: you move the knob up, down, left, or right. They advertise that it’s easier to operate, especially if you need to open it in the dark.

However, it also seems easier for someone else to crack the code.

Assuming a 4-step combination code, how many tries would it take a stranger–at most–to crack the code? (I didn’t read exactly how many steps the code is, I just blindly assumed it would be 4). Assuming it would take 5 seconds to try a given combination, how many minutes, at most, would it take someone to bust open your lock?

You could also consider the following problem: suppose there is no fixed length of the combination. You can make any combination between 1 step long and 4 steps long. In that case, how many possible combos would exist?

Also: see what happens if the combination is 5 steps long.  Or 6 steps long.  The number of possible combos is shooting up.

Can you make a formula for the number of possible combos, if the combination is n steps long?

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