[seqfan] Re: square loop

[Frank Adams-watters]

Robert,

Do you have an easy way to count how many solutions there are of a given size? That would make an interesting sequence.

Franklin T. Adams-Watters


-----Original Message-----
From: Robert Gerbicz <robert.gerbicz at gmail.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Sat, Oct 12, 2019 3:53 pm
Subject: [seqfan] Re: square loop

You're right, last year I've solved it (the two cases) completely, see in:
https://www.mersenneforum.org/showthread.php?t=22915 (see post number 22).

<hv at crypt.org> ezt írta (időpont: 2019. okt. 12., Szo, 22:01):

> After a recent puzzle in New Scientist.
>
> The integers 1 .. 32 can be arranged in a loop such that each consecutive
> pair sums to a square:
>   32 4 21 28 8 1 15 19 26 23 2 14 22 27 9 16
>   20 29 7 18 31 5 11 25 24 12 13 3 6 30 19 17
>
> My trial code to test for this finds n = 32 is the smallest for which this
> is possible, and finds solutions for each of 32 to 44; however the code
> is becoming unusably slow as n increases.
>
> My suspicion is that it is possible precisely for n >= 32, can someone
> prove this, or at least show an upper bound for an n for which the loop
> is not possible?
>
> If we require only a sequence rather than a loop, the first solution
> occurs with n = 15:
>   8 1 15 10 6 3 13 12 4 5 11 14 2 7 9
> .. and it appears there are solutions for n in { 15, 16, 17, 23 } and
> all n >= 25 (tested up to n = 47).
>
> I would guess that the two examples might be of interest in the OEIS, but
> the sets of values of n for which loops or sequences are (or are not)
> possible would not be suitable as OEIS sequences.
>
> Hugo
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Seqfan Mailing list - http://list.seqfan.eu/