Crystal Problem

[rocdocmac]

Has this one appeared before?

[Pickett]

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166,667,166,667,000,000

 

[bonanova]

No, and after playing with it for a few minutes it seems a nice challenge.

Thanks.

[flamebirde]

If calculators can be used, then the problem is simple. Without a program or calculator, though...

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Each individual term is the series from n=0 to some number X of n. So X=3 would give 3+2+1 or six atoms on the third layer. That part's simple enough.

Each layer can also be defined as 1+2+3+... n where n is how many layers down the tetrahedron it is.

The question is how you can add up all the layers. It would be something like (1+2+3+... 1000000)+ (1+2+3+... 999999) +... 1.

Alternatively, it could become 1000000+2(999999)+3(999998)+... 1000000(1); a modified version of the classic "add up all the numbers from one to one hundred" problem.

 

Edited April 18, 2017 by flamebirde
Doubled.

[flamebirde]

If anyone's still working on this, I also have this...

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Here's the series for this problem: the sum of all numbers from 0 to some number k-1 of ((k-n)^2)n where k is the number of layers. So, the sum of all numbers from zero to one million of ((1000000-n)^2)*n is the answer to the problem. But how to do it without a calculator?

 

[flamebirde]

  On 4/19/2017 at 12:41 AM, flamebirde said:

If anyone's still working on this, I also have this...

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Here's the series for this problem: the sum of all numbers from 0 to some number k-1 of ((k-n)^2)n where k is the number of layers. So, the sum of all numbers from zero to one million of ((1000000-n)^2)*n is the answer to the problem. But how to do it without a calculator?

 

Expand  

Nope, I retract that, it should be

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Series from n=0 to k of n(n+1)/2. My friend pointed out to me that it wasn't a modified version of the "add up all the numbers from one to one hundred" problem, it was literally exactly that problem. I still have no idea how I could work it out by hand.

 

[Pickett]

Isn't my answer above correct?

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Isn't it just the tetrahedral numbers...the formula for the series is

S_n = ^n(n+1)(n+2) / ₆

So, for n=1,000,000, you have ^{((1,000,000)(1,000,001)(1,000,002))} / ₆ = ^{1,000,003,000,002,000,000} / ₆ = 166,667,166,667,000,000

 

[Buddyboy3000]

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One way to think about it is to treat the crystal like a pyramid, with some exceptions.

To find the amount of dots on the bottom I did: (l)(h+1)/2 This is like the normal equation except adding the length again makes up for the part of the atoms that are cut off in the straight line. For 1 million, I got (1,000,000)(1,000,001)/2 = 500,000,500,000

With the number of atoms on the bottom, next find the equation to get the answer which is (b)(h+2)/3. Again, the h+2 is to make up for what is cut off. I would include the proof for that but is a bit complicated to explain. 

So, the answer is 500,000,500,000*1,000,002/3 = 166,667,166,667,000,000.

This is the same answer Pickett got, so he got the right answer first.

 

 

Edited April 21, 2017 by Buddyboy3000
Had a sentence repeat itself

[rocdocmac]

Well done all of you! The correct answer is indeed 166 667 166 667 000 000

 

Edited April 22, 2017 by rocdocmac