Math quiz
- Thread starter jazzc
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ShadeDream
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I give up, I can't find the formula as I don't know how to. I managed to find the open lockers but I used excel to do this. When trying to figure out the formula I came across patterns which reminded me of something I did in high school/college but I can't remember what it was.
jazzc
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- Jan 27, 2009
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One more person added to the winners list and it 's time to post the solution. Congratulations to all who participated, I hope you had a lot of fun 
The key to the answer is the question "which persons are going to toggle a given locker?". The answer to this becomes obvious after a bit of trial, for example locker number 8 will be visited only by persons 1,2,4 and 8.
Only the persons with number being a factor of the locker 's number will visit and toggle a given locker! The only way for a locker to end up open is if an odd # of people visit the locker. (because they start closed and if it 's an even # they 'll end up closed again)
The only set of numbers who have an odd # of factors are square numbers (why? because one of the multiplication pairs is composed by the same number e.g. 3x3 -> 1 number).
So, the next question becomes, how many square numbers are there up to 20? That 's easy , you can count them. But how many are there in a billion? That 's also easy! Remember, a sqare number is x^2 (e.g. 16 is 4^2). Well, how many other square numbers exist up to that? Obviously 4 (1^2, 2^2, 3^2, and 4^2)
So, to find how many square numbers there are in 1 billion, we just take the square root of 1 billion, and round it down to the closest integer -> 31622. Indeed, 31622^2 = 999.950.884 while 31623^2 = 1.000.014.129 > 1 billion. So there are 31622 squares (and open lockers) in one billion
The key to the answer is the question "which persons are going to toggle a given locker?". The answer to this becomes obvious after a bit of trial, for example locker number 8 will be visited only by persons 1,2,4 and 8.
Only the persons with number being a factor of the locker 's number will visit and toggle a given locker! The only way for a locker to end up open is if an odd # of people visit the locker. (because they start closed and if it 's an even # they 'll end up closed again)
The only set of numbers who have an odd # of factors are square numbers (why? because one of the multiplication pairs is composed by the same number e.g. 3x3 -> 1 number).
So, the next question becomes, how many square numbers are there up to 20? That 's easy , you can count them. But how many are there in a billion? That 's also easy! Remember, a sqare number is x^2 (e.g. 16 is 4^2). Well, how many other square numbers exist up to that? Obviously 4 (1^2, 2^2, 3^2, and 4^2)
So, to find how many square numbers there are in 1 billion, we just take the square root of 1 billion, and round it down to the closest integer -> 31622. Indeed, 31622^2 = 999.950.884 while 31623^2 = 1.000.014.129 > 1 billion. So there are 31622 squares (and open lockers) in one billion
Last edited:
ShadeDream
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I feel so stupid. ^^
Wilson Grant Fisk
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- Nov 10, 2012
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I was going to give this a go but having read through the comment posts and seeing that people are writing code to solve it I think I will give it a miss. Already starting to get a headache thinking about it.
jazzc
Elite Member
- Jan 27, 2009
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Wilson Grant Fisk
Elite Member
- Nov 10, 2012
- 15,677
- 46,846
That was the solution ?Read the end of page 1 then because I posted the solution there a few minutes ago
That was what gave me a headache !
