Question 1135396: Hi:
My math problem is this: Latisha invited 50 of her closest friends to her birthday, but they decided that 50 presents was too many. Instead, they got 50 boxes and numbered them from 1-50, with presents in only some boxes. When it was time to open gifts, Latisha had to follow certain instructions:
To open every box
Then she had to start with box #2 and close #2, #4, #6, etc. going down the line
Starting with box #3, she was to change every third box, so if it was open she closed it and if it was closed she opened it.
Starting with #4, she had to change every 4th box
Starting with #5, she had to change every 5th box.
She was to go through the line fifty times continuing the pattern.
The open boxes left a pattern, and she had to search those boxes to find it.
Task 1: Which boxes contained gifts? Describe the pattern Latisha found.
The boxes that were open were 1, 4, 9, 16, 25, 36, and 49. The pattern is that the numbers are perfect squares, so 1^2 is 1, 2^ 2 is 4, 3^2 is 9, 4^2 is 16,
5^2 is 25, 6^2 is 36, and 7^2 is 49.
Task two: Explain why you thin this pattern occurred? I don't know how this occured, would it be possible for you to assist me with it? Thank you!
Found 3 solutions by Alan3354, greenestamps, ikleyn:
Answer by Alan3354(69443)   (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
This is a rather clumsy restatement of a classic problem....
After she followed the first instruction -- opening all of the originally closed boxes -- all of the boxes were open. After that, the boxes that were changed an additional even number of times (making the TOTAL number of times changed odd) would end up again open; those that were changed an additional odd number of time would end up closed.
Going through the line 50 times is silly; every time through the line, she starts by opening all the boxes, so at the end of 50 times through the line the result will be the same as after the first time through the line.
The last statement about her having to search the boxes to find the pattern of the open boxes is nonsense. When she has finished following the instructions, it is obvious what the pattern of the open boxes is; no searching is necessary.
So, starting with 50 closed boxes, the boxes that were changed an odd number of times ended up open; those that were changed an even number of times ended up closed. What is it that determines how many times each box is changed? Why are the perfect squares the ones that end up open?
You should discover the reason on your own, rather than having me (or somebody else) tell you. I suggest you choose a few boxes with numbers that are and are not perfect squares and list the numbers for which the state of the box was changed. (I hope it is clear that the divisors of the box number determine when the state of the box will change.) Here are a few....
(1) box #30 -- changed for each divisor of 30: 1 and 30, 2 and 15, 3 and 10, 5 and 6 --> 8 times, an even number; the box will end up closed
(2) box #28 -- changes for each divisor of 28: 1 and 28, 2 and 14, 4 and 7 --> 6 times, an even number; the box will end up closed
(3) box #36 -- changes for each divisor of 36: 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 --> 9 times, an odd number; the box will end up open
Answer by ikleyn(52794)  (Show Source):
You can put this solution on YOUR website! .
See the TWIN problem *) and its numerous solutions at these links
https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.72342.html
https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.93951.html
https://www.algebra.com/algebra/homework/word/mixtures/Mixture_Word_Problems.faq.question.671080.html
https://www.algebra.com/algebra/homework/proportions/Proportions.faq.question.1132246.html (*)
http://mathforum.org/library/drmath/sets/select/dm_1000_lockers.html
I especially recommend you to look into MY SOLUTION at the link marked (*) in the list
https://www.algebra.com/algebra/homework/proportions/Proportions.faq.question.1132246.html
https://www.algebra.com/algebra/homework/proportions/Proportions.faq.question.1132246.html
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*) Strictly saying, the problem I am referring to, is not a TWIN problem - it is a PARENT problem to yours . . .
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