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10 replies to this topic

#1 superprismatic

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Posted 23 September 2011 - 07:31 PM

Here's something I came across in my mathematical perusals:
```0   1   1   2   3   5   8  13 ...
1   3   4   7  11  18  29  47 ...
2   4   6  10  16  26  42  68 ...
...etc...
```
Notice that each row above is fibonacci in that, except for
the first two columns, a number in the ith column is the sum
of the numbers in the (i-1)st and (i-2)nd columns.
Also notice that the first column is just the one-up integers
starting at 0.

Here's how to generate the rest of the array:
Label the columns -1,0,1,2,3,... The first row is the usual
fibonacci sequence. We will genetate the rows consecutively
beginning 1, 2, 3, etc. For the row beginning with N, we
look for N in columns labeled with positive integers and
in the rows above the one we are generating. When we find
N in this manner, we take the number immediately to its right,
add 1 to it, and put this next to N in the line we are generating.
Then we generate the rest of the row using the fibonacci rule

For example, to genetate the row beginning with 1, we find
the 1 in the column labeled 1 and take the 2 immediately to
the right of it, add 1 to it to get 3. That 3 becomes the
number next to the 1 in the line we are generating.

Another example: The row which begins 3. We find a 3 in the
top row with a 5 next to it. We add 1 to the 5 to get the 6
as the element in the 0th column of the row beginning with 3.
Using the fibonacci rule, we find that row to be:
```3   6   9  15  24  39  63 102 ...
```
The claim was made that every positive integer can be found in
one, and only one, positive integer labeled column and
one, and only one, row of this array. Can you prove this?
• 0

#2 phillip1882

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Posted 24 September 2011 - 04:01 AM

took me a bit to understand what you were getting at.
Spoiler for partial solution

Edited by phillip1882, 24 September 2011 - 04:07 AM.

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#3 CaptainEd

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Posted 03 October 2011 - 08:26 PM

Superprismatic, I wanted to post an expression of my pleasure with this puzzle, but I thought I'd wait until I had something constructive to help in its solution. Unfortunately, I can't see deep enough yet to meet the latter goal. So thank you for the puzzle, such a bizarre construction and a fascinating result. (...I'll keep on thinking...)
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#4 superprismatic

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Posted 03 October 2011 - 09:02 PM

Superprismatic, I wanted to post an expression of my pleasure with this puzzle, but I thought I'd wait until I had something constructive to help in its solution. Unfortunately, I can't see deep enough yet to meet the latter goal. So thank you for the puzzle, such a bizarre construction and a fascinating result. (...I'll keep on thinking...)

Join the club. I haven't been able to prove it either. I'm glad you enjoy the puzzle. Thanks for the feedback!
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#5 phillip1882

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Posted 05 October 2011 - 04:45 PM

Spoiler for think i got it

Edited by phillip1882, 05 October 2011 - 04:53 PM.

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#6 CaptainEd

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Posted 05 October 2011 - 05:03 PM

Phillip1882, I can't read your spoiler, but I've got to believe you've not understood the instructions. Here are the first few rows and columns, as I understand the task--12 appears once, 13 appears once, in contrast to your claim:
```  [font=courier new,courier,monospace
-1   0   1   2   3   4   5   6
0   1   1   2   3   5   8  13
1   3   4   7  11  18  29  47
2   4   6  10  16  26  42  68
3   6   9  15  24  39  63 102
4   8  12  20  32  52  84 136
5   9  14  23  37  60  97 157
6  11  17  28  45  73 118 191
7  12  19  31  50  81 131 212
8  13  22  36  58  94 152 246
9  15  25  41  66 107 173 280
10  17  27  44  71 115 186 301
11  19  30  49  79 128 207 335
12  21  33  54  87 141 228 369
13  22  35  57  92 149 241 390   [/font]
```

Edited by CaptainEd, 05 October 2011 - 05:12 PM.

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#7 CaptainEd

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Posted 05 October 2011 - 05:28 PM

Sigh, some days I can't use this editor...
```  -1   0   1   2   3   4   5   6
0   1   1   2   3   5   8  13
1   3   4   7  11  18  29  47
2   4   6  10  16  26  42  68
3   6   9  15  24  39  63 102
4   8  12  20  32  52  84 136
5   9  14  23  37  60  97 157
6  11  17  28  45  73 118 191
7  12  19  31  50  81 131 212
8  13  22  36  58  94 152 246
9  15  25  41  66 107 173 280
10  17  27  44  71 115 186 301
11  19  30  49  79 128 207 335
12  21  33  54  87 141 228 369
13  22  35  57  92 149 241 390[/font]
[font=courier new,courier,monospace]```

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#8 EventHorizon

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Posted 05 October 2011 - 05:33 PM

Spoiler for think i got it

You misunderstood the method of choosing the elements of the column marked '0'
It is not simply the fibonacci numbers plus one.

For instance, the row starting with 4 would require you to find the 4 in the columns with positive integers. It occurs in the column marked '1' and in the row beginning with 1. To the right of that is a 7, so adding 1 to that makes it an 8, not 9 (the fibonacci+1). The next number on that row happens to be the 12 that wouldn't occur anywhere if choosing fib+1.
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#9 phillip1882

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Posted 05 October 2011 - 05:52 PM

alright sorry guys, you're right of course...

Sigh, some days I can't use this editor...

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#10 phillip1882

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Posted 05 October 2011 - 06:16 PM

Spoiler for an interesting observation

Edited by phillip1882, 05 October 2011 - 06:20 PM.

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