20190105, 12:46  #34 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
here's another thing you all may have noticed neither A or B can have more than 2 values mod 4 within them.

20190105, 17:06  #35 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2·2,909 Posts 
Have found plenty of 4+3 solutions but no 4+4 yet. Not sure I can search much further

20190107, 09:57  #36  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2·2,909 Posts 
Quote:
Something I hadn't realized is that if A=[a,b,c,d] and B=[e,f,g,h] is a solution then A=[a1,b1,c1,d1] and B=[e+1,f+1,g+1,h+1] is a solution. I think this means that you can set a=1 and generate all solutions based on this. 

20190107, 10:44  #37 
Jun 2003
11370_{8} Posts 

20190107, 12:47  #38 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2·2,909 Posts 
Unfortunately I have found a bug that invalidates my solutions.
@axn that is potentially a useful idea. 
20190113, 07:46  #39 
Oct 2017
103 Posts 
Eight days ago, I have found a 4+4 solution and a few hours later a 4+5 solution  sufficient for a "*". But the search for a 4+6 solution  [a1,a2,a3,a4] and [b1,b2,b3,b4,b5,b6] with ai+bj = square number – seems to be difficult. Until now I have only three “quasisolutions” with 23 of 24 fulfilled conditions:
cij: = ai +bj = a square number is correct for i = 1 to 4 and j = 1 to 6; only a4 + b6 is no square number. I would like to know which sort of solutions other people with “*” have sent to the puzzle master. Last fiddled with by Dieter on 20190113 at 07:49 
20190113, 17:04  #40 
Jan 2017
3·29 Posts 

20190113, 20:20  #41 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
2^{3}×569 Posts 
The age old binary question: 1 or 0
Just curious if you are having more luck if a1=1 or a1=0?
I found a (4,3) solution quite easily with a1=1. I guess the third option could be: a1 != {0,1} 
20190114, 07:50  #42 
Oct 2017
103 Posts 
[QUOTE=petrw1;505800]Just curious if you are having more luck if a1=1 or a1=0?
I found a (4,3) solution quite easily with a1=1. That's no question for me, because I don't use ai and bj as parameters. My parameters for the search are the square numbers cij = ai + bj. I don't know, if I am allowed to write more here. 
20190114, 11:44  #43 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2×2,909 Posts 
The value for a1 shouldn't make a difference. If you add 1 onto all As and subtract 1 from all Bs then you have a new solution. As such something needs to be fixed. This can be 0 or 1.

20190114, 16:03  #44 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
2^{3}×569 Posts 

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