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# tree pattern puzzle

23 replies to this topic

### #11 curr3nt

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Posted 06 November 2012 - 07:41 PM

He said the 24 on the 4th diagonal was supposed to be black.
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### #12 CaptainEd

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Posted 06 November 2012 - 07:46 PM

Spoiler for White box means
Yes, sorry curr3nt, I was slow catching up to your observation about (4,24), and his corroboration.

Edited by CaptainEd, 06 November 2012 - 07:48 PM.

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### #13 curr3nt

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Posted 06 November 2012 - 08:27 PM

So...

What is special about the following that doesn't apply to anything else?
4: 4, 8, 12, 16, 20, 24
6: 12, 18
8: 4, 8, 12, 16, 20
9:
10:
12: 4, 6, 8, 16
14:
15:
16: 4, 8, 12
18: 6
20: 4, 8

The non primes down the middle are obvious. Why the multples of 4 and 6 for the rest? (Only 4 and 6 for the non-prime diagonals too...)

Edited by curr3nt, 06 November 2012 - 08:28 PM.

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### #14 CaptainEd

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Posted 06 November 2012 - 09:59 PM

Spoiler for Perhaps GCD is key

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### #15 CaptainEd

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Posted 06 November 2012 - 10:11 PM

Spoiler for Simpler statement of the rule

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### #16 curr3nt

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Posted 06 November 2012 - 10:27 PM

Spoiler for I think Captian Ed might have it...

Edited by curr3nt, 06 November 2012 - 10:30 PM.

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### #17 curr3nt

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Posted 06 November 2012 - 10:51 PM

Spoiler for think I fixed it...

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### #18 CaptainEd

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Posted 06 November 2012 - 11:09 PM

If we assume this coloration rule is correct, we are asked to consider what is now the collection of NW-SE diagonals, and answer where there's one with more blacks than reds.

At a snail's pace, I point out that the diagonal that intersects (k,1) consists of all the n1, n2 where n1+n2 = k+1.

And I find it fascinating that there are some that are all white (never mind black and red!) k = 4, 6, 10, 12, 16, for example. There must be an obvious mathematical reason why all pairs summing to those numbers are mutually prime... (blushing) yes, if K is prime, and n1 and n2 add to K, then they don't share any factor, otherwise, K would share that factor.

Edited by CaptainEd, 06 November 2012 - 11:16 PM.

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### #19 curr3nt

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Posted 07 November 2012 - 02:10 PM

Spoiler for first with more Black than Red...

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### #20 phil1882

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Posted 07 November 2012 - 09:43 PM

wow you guys really put a lot of time and effort into this. i appreciate that. the rules are pretty simple, or at least the ones i was trying to follow.
Spoiler for

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