## Problem 210

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Comments, questions and clarifications about PE problems.
viv_ban
Posts: 23
Joined: Mon May 26, 2008 3:09 pm

### problem 210

Can somebody please confirm the answer for N(1000). my answer is 1597548.

quilan
Posts: 182
Joined: Fri Aug 03, 2007 11:08 pm

### Re: problem 210

That answer is (I believe) incorrect. We had a thread discussing this problem a while back, if you'd like to search around for it.
Last edited by quilan on Sun Jan 11, 2009 5:09 pm, edited 1 time in total.
ex ~100%'er... until the gf came along.

daniel.is.fischer
Posts: 2400
Joined: Sun Sep 02, 2007 11:15 pm
Location: Bremen, Germany

### Re: problem 210

All the "Problem xxx" threads (except the newest) have been moved because they cluttered up a lot of subfora, they will return concentrated in one place, but there's some merging and editing work to be done before that.
Il faut respecter la montagne -- c'est pourquoi les gypa&egrave;tes sont l&agrave;.

TripleM
Posts: 382
Joined: Fri Sep 12, 2008 3:31 am

### Re: problem 210

Close, but your answer is too low.

viv_ban
Posts: 23
Joined: Mon May 26, 2008 3:09 pm

### Re: problem 210

I have checked my program and I found a small error, but still I am unable to get the correct answer. Can someone please tell me if I am correct this time.
N(1000) = 1597880
N(10000) = 159814790
N(100000) = 15981722482
N(1000000) = 1598174519142

daniel.is.fischer
Posts: 2400
Joined: Sun Sep 02, 2007 11:15 pm
Location: Bremen, Germany

### Re: Problem 210

Yep.
Il faut respecter la montagne -- c'est pourquoi les gypa&egrave;tes sont l&agrave;.

sinan
Posts: 16
Joined: Mon Sep 15, 2008 10:14 am

### Re: Problem 210

What does |x| + |y| ≤ r mean? It should define a diamond shaped area, right? I don't understand what kind of trap did you fall into? I have found a closed formula that seems to work for small number without writing any code, then what happens? I have a feeling from the messages in this thread that there is a circle thing here but I cannot understand how a diamond shaped area becomes circle??

******* 4
******4 3 4
****4 3 2 3 4
**4 3 2 1 2 3 4
4 3 2 1 0 1 2 3 4
**4 3 2 1 2 3 4
****4 3 2 3 4
******4 3 4
********4

quilan
Posts: 182
Joined: Fri Aug 03, 2007 11:08 pm

### Re: Problem 210

Well, you're absolutely right re: diamond, but you'll see that something subtle happens with regards to forming the triangle. I don't want to take away the joy of discovery from you, but the smallest example that shows the errant behavior starts @ r=24. Do a brute-force of the coordinate space & compare to the correct answer. This problem is quite nice in the way it sneaks that one in there.

Edit: Correct answer is 916 I think... provided my old code still works. And I -think- it was 24 that causes the odd behavior. Might have been 20 or something.
Last edited by quilan on Wed May 06, 2009 4:52 pm, edited 2 times in total.
ex ~100%'er... until the gf came along.

sinan
Posts: 16
Joined: Mon Sep 15, 2008 10:14 am

### Re: Problem 210

quilan wrote:Well, you're absolutely right re: diamond, but you'll see that something subtle happens with regards to forming the triangle. I don't want to take away the joy of discovery from you, but the smallest example that shows the errant behavior starts @ r=24. Do a brute-force of the coordinate space & compare to the correct answer. This problem is quite nice in the way it sneaks that one in there.
r=24. Hmmm. OK I will try to see what happens there.

sinan
Posts: 16
Joined: Mon Sep 15, 2008 10:14 am

### Re: Problem 210

sinan wrote:
quilan wrote:Well, you're absolutely right re: diamond, but you'll see that something subtle happens with regards to forming the triangle. I don't want to take away the joy of discovery from you, but the smallest example that shows the errant behavior starts @ r=24. Do a brute-force of the coordinate space & compare to the correct answer. This problem is quite nice in the way it sneaks that one in there.
r=24. Hmmm. OK I will try to see what happens there.
I get 904 for both. I counted the number of stars in the following.

Code: Select all

                        *
***
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4************
432************
43210************
4321098************
432109876************
43210987654************
4321098765432***********4
**2109876543212*********2**
****0987654321012*******0****
******8765432109012*****8******
********6543210989012***6********
**********4321098789012*4**********
************2109876*****2************
**************09876*****0*2************
****************8765****8**12************
******************654***6***012************
********************43**4****9012************
**********************2*2*****89012************
************************0*****6789012************
**********************2*2345678901234**********
********************4***4567890123456********
******************6*****6789012345678******
****************8*******8901234567890****
**************0*********0123456789012**
************2***********2345678901234
**********4*************45678901234
********6***************678901234
******8*****************8901234
****0*******************01234
**2*********************234
4***********************4
***********************
*********************
*******************
*****************
***************
*************
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***
*


MaJJ
Posts: 49
Joined: Tue Oct 14, 2008 12:14 am

### Re: Problem 210

Hi,

I just want to assure myself that the circle-like shape is not a circle (which I thought when I looked at
Expand
- I thought it's just some rounding errors or I don't know what)... But at a bigger numbers it starts to get
Expand
- and Mathematica probably shouldn't make any rounding errors as long as I'm using the fractional notation. So could anybody just assure me in this? I wasted lots of time making equations for circle, just to see that they don't work...

daniel.is.fischer
Posts: 2400
Joined: Sun Sep 02, 2007 11:15 pm
Location: Bremen, Germany

### Re: Problem 210

I suggest you look at Thales' theorem (or its generalisation). That should tell you exactly what shape the figure is.
Il faut respecter la montagne -- c'est pourquoi les gypa&egrave;tes sont l&agrave;.

MaJJ
Posts: 49
Joined: Tue Oct 14, 2008 12:14 am

### Re: Problem 210

Thales' theorem somehow helped, thanks. But in the end I found out the shape with simply plotting the N(200). Then one little sqrt(2) and I was ready to analyze further

ATM my code works, but it's quite slow. And I think I have a nice way to calculate it! But still, the more N, the more loops

Code: Select all

Timing[PE210[1000]]
{0.297, 1597880}
Timing[PE210[2000]]
{1.14, 6392158}
Timing[PE210[3000]]
{2.61, 14382796}
I guess I'm not going to reach the 1,000,000,000 this way...
Is there a solution which doesn't require loops? Or at least not so many loops? (for 1000 my code does cca 1650 loops - I can PM my algo if you want)

EDIT: I think I just found a way to make it 4x faster, but I'll have to rewrite it into the code... But even if 4x faster, it'll run for eternity...

thedoctar
Posts: 74
Joined: Fri Apr 15, 2011 11:57 am
Location: Sydney, Australia

### Re: Problem 210

N(10^7)=removed
N(10^8)=removed

Is this correct?
Last edited by thedoctar on Mon Jul 23, 2012 10:53 am, edited 1 time in total.
4x Intel(R) Core(TM) i3-2330M CPU @ 2.20GHz

fabas indulcet fames

hk
Posts: 10871
Joined: Sun Mar 26, 2006 10:34 am
Location: Haren, Netherlands

### Re: Problem 210

[mar]don't post intermediate results[/mar]

You could as well calculate your answer for 10^9 and use the PE answer box couldn't you?

thedoctar
Posts: 74
Joined: Fri Apr 15, 2011 11:57 am
Location: Sydney, Australia

### Re: problem 210

viv_ban wrote:I have checked my program and I found a small error, but still I am unable to get the correct answer. Can someone please tell me if I am correct this time.
N(1000) = 1597880
N(10000) = 159814790
N(100000) = 15981722482
N(1000000) = 1598174519142
I get the same results as this person and daniel.is.fisher confirms them in the next post, but my answer for 10**9 is wrong, which is why I posted my higher results. Could you just double check if the above results are correct? Or could you PM me the answer for 10**7?
4x Intel(R) Core(TM) i3-2330M CPU @ 2.20GHz

fabas indulcet fames

hk
Posts: 10871
Joined: Sun Mar 26, 2006 10:34 am
Location: Haren, Netherlands

### Re: Problem 210

viv_bans values are correct.
BTW daniel.is.fisher was one of the team that were involved in the development of this problem so you should not doubt him on this.
I'm suspecting you have an accuracy problem about which already much has been written in this thread.

thedoctar
Posts: 74
Joined: Fri Apr 15, 2011 11:57 am
Location: Sydney, Australia

### Re: Problem 210

Well I'm sure he was, however I just wanted to be absolutely sure, as he might've misread the numbers or something similar, and although I know it's unlikely, I just wanted to be absolutely sure. I also thought that a problem with floating point precision would've come up at 10^6, as it is a pretty large value, but then again I wouldn't know. I'll try some of the methods in this thread to see if I get the correct answer.

Thanks for your help.

EDIT: yep, you were right! I got the answer right. Wow, I thought that floor(sqrt(x)) would give you an accurate floor of the root of x, but apparently not! Thanks again!!
4x Intel(R) Core(TM) i3-2330M CPU @ 2.20GHz

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coreyjkelly
Posts: 7
Joined: Fri Dec 28, 2012 5:54 am

### Re: Problem 210

I've been banging my head on my desk all night because of this one. Could somebody confirm that Python isn't plagued with the accuracy and overflow issues discussed here? All of my testing would indicate that the large numbers are properly dealt with, and math.floor(x**0.5) seems to function properly.

My results for values up to r=10^6 were consistently off by 1, which I couldn't justify, but adding 1 to my answer for 10^9 doesn't work

thundre
Posts: 356
Joined: Sun Mar 27, 2011 10:01 am

### Re: Problem 210

coreyjkelly wrote:All of my testing would indicate that the large numbers are properly dealt with, and math.floor(x**0.5) seems to function properly.
I don't know that much about Python. I gather that its integers are naturally "big". But how would the interpreter know how much precision you need in an exponentiation?