From my site, a few years ago ... At the dinner, I met Bernardo Recaman Santos of Columbia. He gave me a great puzzle: Arrange the numbers 1-15 in a sequence with the property that every two consectutive numbers sum to a square number. http://www.mathpuzzle.com/WPC2000.html
So ... I give credit to Bernardo Recaman Santos. --Ed Pegg Jr, www.mathpuzzle.com --- Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
Can anyone provide any reference to the following problem?
For which n is is possible to arrange the numbers from 1 to n in a chain so that the sums of adjacent links are always squares?
Partial answer: n = 15, 16, 17:
(16-)9-7-2-14-11-5-4-12-13-3-6-10-15-1-8(-17)
Elwyn Berlekamp & I got it from Yoshiyuki Kotani in Edmonton last July, who was examining the corresponding problem with cubes in place of squares, and said that he had seen the squares problem somewhere. I think the answer for any power is `for all sufficiently large n'. We proposed the corresponding problem with Fibonacci numbers in place of squares and have the theorem:
There is a chain with adjacent links adding to Fibonacci numbers just if n = 9, 11 or F_k or F_k - 1 where F_k is a Fibonacci number with k > 3.
Before publishing formally, we'd like to trace the origins, if we can. R.
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