Jump to content
BrainDen.com - Brain Teasers
  • 0

tree pattern puzzle



23 answers to this question

Recommended Posts

  • 0

congrats current. that is indeed the next four rows.

any clue on my second question? I'm honestly not sure myself.

there actually was an outer most layer to the picture that i deliberately removed to make the puzzle more challenging.

the picture is based on the primes, or more specifically the prime factorization.


Edited by phil1882
Link to comment
Share on other sites

  • 0

How are you determining red and black?

I see the rows for the primes where red is flagged when the mod is equal to 0. So I was using red when n1 mod n2 = 0 and black when n1 and n2 share a factor but that gives me the reverse for the non-primes. I also had to use red if prime and black if not when n1 = n2.

I get the following which matches the pattern of squares but not the coloring...


Link to comment
Share on other sites

  • 0

wait... in your triangle the "12th" diagonal has a square on 2,3,4,6,8 and 10 but not 9?

Another apparent oddity: the "diagonal" for 4 has a BLACK square for 4, 8, 12, 16, 20, but RED for 24

ALso, the "diagonal" for 9 has Black for 9, but Red for 18

Of course, these wouldn't be oddities if we understood the pattern...but I don't :wacko:

Link to comment
Share on other sites

  • 0

It appears that there's a white box (ie. no box at all) for n1, n2 exactly when n1 and n2 are relatively prime. EXCEPT for the white boxes on (9,12) as curr3nt pointed out, and (10,15).

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

Edited by CaptainEd
Link to comment
Share on other sites

  • 0


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



12: 4, 6, 8, 16



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
Link to comment
Share on other sites

  • 0

Here's a rule, and a few exceptions. Since the OP is acknowledged to have an error, maybe the exceptions are errors as well.

Rules, applied sequentially until a rule applies: cell (n1, n2) is---

* white if n1 and n2 are relatively prime, else

* red if n1 is prime or n2 is prime, else

* black if n1 divides n2 or n2 divides n1, else

* red if GCD (n1, n2) is prime, else

* black


* (4,24) is red, should be black (acknowledged by poster)

* (10,15) is white, should be red

* (9,12) is white, should be red

* (9,18) is red, should be black

Link to comment
Share on other sites

  • 0

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
Link to comment
Share on other sites

  • 0


13 Black vs 10 Red vs 10 White

Black: (first 6 counted twice for reverse)








Red: (Counted twice for reverse)












Other numbers with more black than red:

58: 23 vs 20

70: 27 vs 20

82: 31 vs 28

98: 31 vs 28

106: 41 vs 30

118: 47 vs 40

130: 47 vs 44

138: 47 vs 44

142: 55 vs 40

154: 55 vs 52

166: 63 vs 56

178: 79 vs 52

194: 57 vs 54

198: 63 vs 56

202: 71 vs 68

214: 83 vs 60

218: 71 vs 68

226: 79 vs 76

238: 95 vs 80

250: 107 vs 72

258: 83 vs 80

262: 95 vs 88

274: 95 vs 92

278: 95 vs 88

286: 111 vs 80

298: 123 vs 96

Link to comment
Share on other sites

  • 0

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.

take the horizontal pattern, and repeat it diagonally, starting at 2. note that the white space is cut in half. if a number is all white horizontally, then the outer most pixel should be red, else black.

Link to comment
Share on other sites

  • 0

I've got to confess that I couldn't relate what you just said to the analysis curr3nt and I have been doing. I fear we are off target.

Here's a question. Did you originate this problem? (For example, when curr3nt spotted the surprising color on 4x24, you agreed that it should have been black. That makes me think you created this.) Or did you find it elsewhere (for example, you said you had removed the outer shell. That made me think someone else originated it).

Link to comment
Share on other sites

  • 0

Thank you, phil1882, for finding and extending such an interesting puzzle! Now that I view the video a few times, I understand what you were saying about how the diagram is created. Overcoming my disbelief that such a simple mechanism could create a chart representing presence or absence of common factors was a pleasurable and enlightening experience.

Thank you, curr3nt, for attacking this puzzle and sharing your progress with us.

Link to comment
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.

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.


  • Recently Browsing   0 members

    • No registered users viewing this page.
  • Create New...