Candles....(From CMC test)
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- bristy1588
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Noushin has $n$ candles. On the first day she lights 1 candle for 1 hour, on the second day she lights 2 candles for 2 hours.... Like this on the n-th day she lights $n$ candles for $n$ hours. The candles chosen according to one's wish. If after the n-th day all the candles had been lighted for the same number of hours, find $n$.
Bristy Sikder
Re: Candles....(From CMC test)
candles can be lit maximum :n(n+1)/2 =m hr.m must be an intreger.so n must be odd.
nafisa
- bristy1588
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Re: Candles....(From CMC test)
Confused, i meant on day 2, 2 candles were lit each for 2 hrs.. and so on
Bristy Sikder
- nafistiham
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Re: Candles....(From CMC test)
যেকোনো বেজোড় সংখ্যাই এমন হতে পারে
ধরা যাক,$n=2k+1$
$1+2+3+...........+2k+(2k+1)$
এখন, $1+2k=2+2k-1=3+2k-3=.............$
সুতরাং, এভাবে মোম ব্যাবহার করা যেতে পারে ।
ধরা যাক,$n=2k+1$
$1+2+3+...........+2k+(2k+1)$
এখন, $1+2k=2+2k-1=3+2k-3=.............$
সুতরাং, এভাবে মোম ব্যাবহার করা যেতে পারে ।
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
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- bristy1588
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Re: Candles....(From CMC test)
Tiham,
Lets see the case when n=3. The 3 candles are A,B and C. Now. We know on the first day 1 candle was lit, candle A, therefore, After day 1, Candle A was lit for 1 hour and the candle B,C for 0 hours, Now let on day 2, candles B and C were lit, each for 2 hours. So after day 2 in total, Candle A was lit for 1 hour, Candle B for 2 hours and Candle C for 3 hours. At day 3, All the 3 candles were lit for 3 hours each. In total after day 3, A was lit for 4 hours, B for 5 hours and C for 5 hours. However, all candles were supposed to be lit for the same number of hours.
I think u did not understand the question.
Lets see the case when n=3. The 3 candles are A,B and C. Now. We know on the first day 1 candle was lit, candle A, therefore, After day 1, Candle A was lit for 1 hour and the candle B,C for 0 hours, Now let on day 2, candles B and C were lit, each for 2 hours. So after day 2 in total, Candle A was lit for 1 hour, Candle B for 2 hours and Candle C for 3 hours. At day 3, All the 3 candles were lit for 3 hours each. In total after day 3, A was lit for 4 hours, B for 5 hours and C for 5 hours. However, all candles were supposed to be lit for the same number of hours.
I think u did not understand the question.
Bristy Sikder
Re: Candles....(From CMC test)
Cracked it...
Last edited by Labib on Sun Dec 18, 2011 3:34 pm, edited 2 times in total.
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- bristy1588
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Re: Candles....(From CMC test)
Labib,
Please tell me where I m getting u wrong:
Let me denote the total number of hours that each candle was lit for after $n$ days as $K$. And let the candle which was not lit on the $(n-1)$ is Candle A.
On the $(n-2)$ day, We have 2 cases:
1. Candle A was lit:
In this case, On the $(n-3)$ day, Candle A was lit:$(k-n-(n-2))$. 2 candle was lit for $(k-n-(n-2))$ hrs and $(n-3)$ candles were lit for$(k-n-(n-1)-(n-2))$ hrs.
2. Candle A was NOT lit:
In this case, On the $(n-3)$ day, Candle A was lit for $(k-n)$ hrs. 1 candle was lit for $(k-n-(n-1))$ hrs and $(n-3)$ candles were lit for $(k-n-(n-1)-(n-2))$ hours
In Either Way, I do not get what u proposed to happen on the $(n-3)$ day. Could u please explain?
Please tell me where I m getting u wrong:
Let me denote the total number of hours that each candle was lit for after $n$ days as $K$. And let the candle which was not lit on the $(n-1)$ is Candle A.
On the $(n-2)$ day, We have 2 cases:
1. Candle A was lit:
In this case, On the $(n-3)$ day, Candle A was lit:$(k-n-(n-2))$. 2 candle was lit for $(k-n-(n-2))$ hrs and $(n-3)$ candles were lit for$(k-n-(n-1)-(n-2))$ hrs.
2. Candle A was NOT lit:
In this case, On the $(n-3)$ day, Candle A was lit for $(k-n)$ hrs. 1 candle was lit for $(k-n-(n-1))$ hrs and $(n-3)$ candles were lit for $(k-n-(n-1)-(n-2))$ hours
In Either Way, I do not get what u proposed to happen on the $(n-3)$ day. Could u please explain?
Bristy Sikder
Re: Candles....(From CMC test)
Fixed the bug... It should be okay... I think...
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
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"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
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- bristy1588
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- Joined:Sun Jun 19, 2011 10:31 am
Re: Candles....(From CMC test)
Labib,
Why didnt u consider the case 1. i.e When Candle$A$ was lit. How do u prove that Candle A was never lit on day $(n-2)$ ?
Why didnt u consider the case 1. i.e When Candle$A$ was lit. How do u prove that Candle A was never lit on day $(n-2)$ ?
Bristy Sikder
Re: Candles....(From CMC test)
Then I think the solution is wrong...
Will try to think on it tomorrow.
Will try to think on it tomorrow.
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
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"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes