Problem

[Richard Guy]

The square bracelet conjecture is answered
in a preprint of Berlekamp & Guy.  We should
make this more available.

On Sun, 22 Jan 2006, Roberto Tauraso wrote:

> This goes in the direction of a similar problem involving primes
> that you can find in JIS:
> http://www.cs.uwaterloo.ca/journals/JIS/green.html
> Do you think that your greedy algorithn is actually
> a proof by induction?
> The problem with squares is connected with a much harder
> conjecture: there exists a square loop (circular permutation
> of the numbers 1 to n such that the sum of any two consecutive
> numbers is a square) for every n>=32 (see A071984).
> Of course for even n the loop gives also a pairing.
>
> Roberto Tauraso
>
> On Sun, 22 Jan 2006, Richard Guy wrote:
>
>> I believe that the following greedy algorithm
>> works for all sufficiently large  n (>31?)
>>
>> Take largest odd square < 4n
>>
>> e.g. n=101, 361=202+159=...=181+180,
>>
>> reducing the problem to  2n=158
>>
>> n = 79:  289=158+131=...=145+144
>> n = 65:  225=130+95=...=113+112
>> n = 47:  169=94+75=...=85+84
>> n = 37:  121=74+47=...=61+60
>> n = 23:   81=46+35=...=41+40
>> n = 17:   49=34+15=...=25+24
>> n = 7:    25=14+11=13+12
>> n = 5  can't be done, but  n = 7
>> can be a different way (see below)
>>
>> I don't think that there are any descents
>> much worse than
>>
>> 94, 86, 58, 54, 30, 10? and
>> 93, 87, 57, 55, 29, 11?
>>
>> and presumably 29 and 30 can be done
>> a different way.
>>
>> Here's a particularly happy one:
>>
>> 100, 80, 64, 48, 36, 24, 16, 8, 4.
>>
>> R.
>>
>> On Sun, 22 Jan 2006, Roberto Tauraso wrote:
>>
>>> Here there are some matchings:
>>> n=4:
>>> 8-1 7-2 6-3 5-4
>>> n=7:
>>> 14-2 13-3 12-4  11-5 10-6 9-7
>>> 8-1
>>> n=8:
>>> 16-9 15-10 14-11 13-12
>>> 8-1 7-2 6-3 5-4
>>> n=9:
>>> 18-7 17-8 16-9
>>> 15-1 14-2 13-3 12-4 11-5 10-6
>>> n=12:
>>> 24-1 23-2 22-3 21-4 20-5 19-6 18-7 17-8 16-9 15-10 14-11 13-12
>>> n=13:
>>> 26-10 25-11 24-12 23-13
>>> 22-3 21-4 20-5 19-6 18-7 17-8 16-9
>>> 15-1 14-2
>>> n=14:
>>> 28-21
>>> 27-9 26-10 25-11 24-12 22-14 20-16
>>> 23-2 19-6 18-7 17-8
>>> 15-1 13-3
>>> 5-4
>>> n=15:
>>> 30-19 29-20 28-21 27-22 26-23
>>> 25-11
>>> 24-1 18-7 17-8 16-9 15-10 13-12
>>> 14-2
>>> 6-3 5-4
>>> n=16:
>>> 32-17 31-18 30-19
>>> 29-7 28-8 27-9 26-10 25-24
>>> 22-14 20-16
>>> 23-2 21-4
>>> 15-1 13-12 11-5
>>> 6-3
>>>
>>> Bye,
>>> Roberto Tauraso
>>>
>>
>