In my archives, I found an extension .... Joe Kisenwether has upped the ante. Arrange the numbers 1-32 in a circle so that any two adjacent numbers sum to a square number. The answer is unique. A nice picture of the answer is http://www.mathpuzzle.com/sqcircle.gif So ... Bernardo Recaman did the original problem, and Joe Kisenwether extended it. Nob Yoshigahara was the first person I know of to ask the corresponding question for cubes. --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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