Boxes and Cards Problem

For students of class 9-10 (age 14-16)
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Tasnood
Posts: 73
Joined: Tue Jan 06, 2015 1:46 pm

Boxes and Cards Problem

Unread post by Tasnood » Fri Mar 02, 2018 11:05 pm

There are $ 7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box $\fbox {1}$ , the second in box $\fbox{2}$ , and so forth up to the seventh card which is placed in box $\fbox{7}$ . You then start working back in the other direction, placing the eighth card in box $\fbox{6}$ , the ninth in box $\fbox{5}$ , up to the thirteenth card being placed in box $\fbox{1}$ . The fourteenth card is then placed in box $\fbox{2}$ , and this continues until every card is distributed. What box will the last card be placed in?

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samiul_samin
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Joined: Sat Dec 09, 2017 1:32 pm

Re: Boxes and Cards Problem

Unread post by samiul_samin » Sat Mar 03, 2018 12:25 am

Answer
Box No.$ 3$
Solution
Note that
$2015=2016-1=(12×168)-1$,
So,2015 is in form $12k-1$ where $k=1,2,3,...$
We are placing cards in the boxes in such a manner that every 12th card will be plced at the same box....($1$)
Now,We have to put card no.$11$ in the $3$rd box(According to the statement of question).It is in $12k-1$ form.If we follow ...($1$),we can say that we will get every $12k-1$ type numbers written cards in the $3$rd box.
So,we are done.
I think it is correct if I have got the question correctly.It is an interesting problem.

Mathlomaniac
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Joined: Wed Mar 07, 2018 11:35 pm
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Re: Boxes and Cards Problem

Unread post by Mathlomaniac » Thu Mar 08, 2018 10:36 pm

samiul_samin wrote:
Sat Mar 03, 2018 12:25 am
Answer
Box No.$ 3$
Solution
Note that
$2015=2016-1=(12×168)-1$,
So,2015 is in form $12k-1$ where $k=1,2,3,...$
We are placing cards in the boxes in such a manner that every 12th card will be plced at the same box....($1$)
Now,We have to put card no.$11$ in the $3$rd box(According to the statement of question).It is in $12k-1$ form.If we follow ...($1$),we can say that we will get every $12k-1$ type numbers written cards in the $3$rd box.
So,we are done.
I think it is correct if I have got the question correctly.It is an interesting problem.

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