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December 22nd, 2012, 06:12 AM   #1
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please find minimal n







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December 22nd, 2012, 08:33 AM   #2
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Re: please find minimal n

Why have you not just done the arithmetic? If you have a calculator this is easy. And apparently you have a computer! If you are running windows, you can use the calculator that comes with it: square root of 2 is about 1.414214 and square root of 3 is about 1.732051 so that . That to the 6 th power is a little bit larger than 969 so the smallest integer larger than that is 970.
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December 22nd, 2012, 07:11 PM   #3
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Re: please find minimal n

Quote:
HallsofIvy wrote : Why have you not just done the arithmetic? If you have a calculator this is easy. And apparently you have a computer!
sorry! no use of calculator or computer.
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December 22nd, 2012, 07:20 PM   #4
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Re: please find minimal n

Wierd that root 2 plus root 3 gives pi correct to 2 decimal places...

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December 22nd, 2012, 08:04 PM   #5
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Re: please find minimal n

If we use the binomial theorem we only have 1 difficult calculation, root 6.







Up to this point, no messy arithmetic involved, just integer addition and integer multiplication.

Now, there are many easy ways to extract the root of 6 manually to 5 or 6 decimal places and then multiply by 198, i used 198(2.4494 = 484.997. Since this is close to 485 i rounded the 8 to 9 and used 2.44949 and since i stayed under 485 i knew i was safe...

485 + 484.997 = 969.997 so 970 is the answer.

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December 22nd, 2012, 09:11 PM   #6
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Re: please find minimal n

a=sqrt(2):b=sqrt(3)

((a+b)^3)^2 = (11a + 9b)^2 = 485 + 198ab = 485 + 198SQRT(6) (YEA...as per our favorite Agent!)

n = 485 + 198SQRT(6)
(n - 485) / 198 = SQRT(6)

So , knowing 3 > SQRT(6) > 2, then 1079 > n > 881;
so Albert must have a "trick of his own" to wrap that up cleanly...
(even if agentredlum's is accurate...but "seems" to need a calculator a bit)
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December 23rd, 2012, 01:06 AM   #7
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Re: please find minimal n

and 24 < sqrt(600) < 25 so 2.4 < sqrt(6) < 2.5
Now, (a + 1)^2 - a^2 ~= 2a and
60000 - 57600 = 2400 = 2a*m where a = 240 so m, = 5 and
sqrt(60000) ~= 245.
245^2 = 240 * 250 + 25 > 60000 so 244 < sqrt(60000) < 245.
Gives 2.44 * 198 = 2.44 * (200 - 2) = 488 - 4.88 = 483.12
2.45 * 198 = (2.44 + 0.01) * 198 = 483.12 + 1.98 = 485.1

Yields
485 + 483.12 = 968.12 < 485 + 198sqrt(6) = (sqrt(2) + sqrt(3))^6 < 485 + 485.1 = 970.1

And n = 969 or 970 without calculator.
Since 240 * 250 + 25 = 60025 is slightly more than 60000, one might argue that sqrt(60000) > 244.7 and so
198 * sqrt(6) > 198 * 2.447 = 198 * (2.44 + 0.007) = 483.12 [from above] + (200 - 2) * 0.007 = 483.12 + 1.4 - 0.014 > 484 so n > 969 which implies n = 970
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December 23rd, 2012, 01:14 AM   #8
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Re: please find minimal n

keep trying ! I will post a very tricky method ! (not mine) soon
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December 23rd, 2012, 03:02 AM   #9
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Re: please find minimal n

using binomial expansion :


we get the minimal n=970
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December 23rd, 2012, 06:56 AM   #10
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Re: please find minimal n

Quote:
Originally Posted by Denis
(YEA...as per our favorite Agent!)
I hope all your dreams come true this holiday season, even if you are just an imagination of my figment...
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