Jump to content
BrainDen.com - Brain Teasers
  • 0
Sign in to follow this  
rocdocmac

Crystal Problem

Question

8 answers to this question

Recommended Posts

  • 0

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

Spoiler

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 by flamebirde
Doubled.

Share this post


Link to post
Share on other sites
  • 0

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

Spoiler

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?

 

Share this post


Link to post
Share on other sites
  • 0
On 4/18/2017 at 5:41 PM, flamebirde said:

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

  Reveal hidden contents

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?

 

Nope, I retract that, it should be

Spoiler

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.

 

Share this post


Link to post
Share on other sites
  • 0

Isn't my answer above correct?

Spoiler

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

Sn = 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

 

Share this post


Link to post
Share on other sites
  • 0
Spoiler

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. :unsure:

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 by Buddyboy3000
Had a sentence repeat itself

Share this post


Link to post
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Answer this question...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
Sign in to follow this  

  • Recently Browsing   0 members

    No registered users viewing this page.

×
×
  • Create New...