21 Jan 2013

Math-e-Monday: Prime Combination Lock

Derek has just bought himself a smart new padlock with a combination lock. It is one of those with a 3-digit code that you enter by turning each of the 3 wheels. Derek is playing with it, trying to think of a really clever number that he wouldn’t forget.

He notices that the code was set to 200 and decides to play a game. He picks the middle and right wheels; clicking one of them up a number and the other down a number he gets 291. He continues this until he gets back to 200. Then he tries the same thing with the middle and left wheels, and finally with the left and right wheels.

Derek smiles to himself

a)    Write down the prime numbers that Derek finds using his method.


Derek then wants to find out if he can make every possible 3-digit number. He still starts at 200, turns one wheel up and one wheel down, then writes his new number down. But in his next step he can choose any pair of wheels he wishes, not just the two he started with. At each step he writes down his number so as to keep track.



b)    Write down any new prime numbers that Derek finds.

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Part (a) is the type of question found in some mathematics competitions. Such tests are usually done without any electronic assistance such as a calculator or computer. Try this just with pen and paper. If you figure out the pattern, this should take about 3 minutes.

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Part (b) would require an astonishing memory for prime numbers! Given a table of primes, an efficient algorithm would still yield a fairly quick solution without electronic aids.

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Part (c). Was there no part (c)? There is now! Can you generalize these results starting with a padlock set to N00, where N is between 0 and 9 inclusive? We’ve done N=2 already.

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