Problem of the Week #59: Monday August 19th, 2024
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by this twitter post.
There’s a combination lock with a 3 digit code with no repeated digits, and for each of our attempts to guess the code it tells us the number of correct digits and ones in the correct places.
a). How many tries do we need to guarantee finding the right combination?
b). Does it take more tries to guarantee guessing a 2 digit code or a 3 digit code?
c). How many more tries do we need to guarantee guessing a 3 digit code if repeated digits are allowed?
d). How many tries do we need to guarantee finding the right combination for a 4 digit code if repeated digits are allowed?
e). If the first two tries to a 4 digit combination lock with no repeated digits give these results, what combination(s) should we try next?
f). If the combination lock accepts passwords up to 4 digits long, and repeated digits are allowed, how many tries do we need to guarantee guessing the right combination?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.