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Fibonacci appears again (Posted on 2015-08-21) Difficulty: 3 of 5
| 0  0  1  0  0|
| 0  1  0  1  0|
| 1  0  2  0  1|
| 1  3  0  3  1|
| 4  1  6  1  4|
| 5 10  2 10  5|
|15  7 20  7 15|
|22 35 14 35 22|
...
Once you've figured out how to keep adding rows, prove the following:

The absolute difference between the numbers on any two (non equal) columns is the Fibonacci sequence.

  Submitted by Jer    
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Solution: (Hide)
Number the columns and put variables in them
(3)(2)(1)(2)(3) 
a b c b a

I will consider only the difference (1-2) since the others work the same
where b ≤ a < c and c-b > a-b, (1-2)=c-b next row is
a+b a+c 2b a+c a+b
(1-2)=a+c-2b
next row is
2a+b+c a+3b 2a+2c a+3b 2a+b+c
(1-2)=a-3b+2c

(1-2) in the problem clearly starts 1,1,2,3... as the Fibonacci sequence does.
(c-b)+(a-2b+c)=(a-3b+2c)
What I showed above is that each difference is the sum of the previous two differences.

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